Here is the doctrine of Sola Fide as defined by the classic Reformed statement of faith, the Westminster Confession of Faith, Chapter 11, Section 1:
Those whom God effectually calleth, He also freely justifieth: not by infusing righteousness into them, but by pardoning their sins, and by accounting and accepting their persons as righteous, not for anything wrought in them, or done by them, but for Christ’s sake alone; nor by imputing faith itself, the act of believing, or any other evangelical obedience to them, as their righteousness, but by imputing the obedience and satisfaction of Christ unto them, they receiving and resting on Him and His righteousness by faith; which faith they have not of themselves, it is the gift of God.
Here is a definition of justification from the Catechism of the Catholic Church, #1989-1992 (footnotes removed):
1989 The first work of the grace of the Holy Spirit is conversion, effecting justification in accordance with Jesus' proclamation at the beginning of the Gospel: "Repent, for the kingdom of heaven is at hand." Moved by grace, man turns toward God and away from sin, thus accepting forgiveness and righteousness from on high. "Justification is not only the remission of sins, but also the sanctification and renewal of the interior man.
1990 Justification detaches man from sin which contradicts the love of God, and purifies his heart of sin. Justification follows upon God's merciful initiative of offering forgiveness. It reconciles man with God. It frees from the enslavement to sin, and it heals.
1991 Justification is at the same time the acceptance of God's righteousness through faith in Jesus Christ. Righteousness (or "justice") here means the rectitude of divine love. With justification, faith, hope, and charity are poured into our hearts, and obedience to the divine will is granted us.
1992 Justification has been merited for us by the Passion of Christ who offered himself on the cross as a living victim, holy and pleasing to God, and whose blood has become the instrument of atonement for the sins of all men. Justification is conferred in Baptism, the sacrament of faith. It conforms us to the righteousness of God, who makes us inwardly just by the power of his mercy. Its purpose is the glory of God and of Christ, and the gift of eternal life:
The Protestant doctrine of justification, Sola Fide, teaches that we are made acceptable to God solely because the righteousness of Christ is imputed to us. Nothing at all can be added to this, nor can anything else make us acceptable to God. Protestants believe that we are made inwardly holy in the process of sanctification, where the Holy Spirit conforms us in our inward lives and behaviors to the image of Christ, as a fruit of justification, but it is the imputation of Christ's righteousness alone (that is, the crediting of his righteousness and satisfaction to our account) that makes us acceptable to God.
So, is Sola Fide compatible with Catholic doctrine? It depends on how we interpret certain aspects of it. In particular, it greatly depends on how those who hold Sola Fide answer this question: "How does sanctification fit into our relationship with God? When you say we are acceptable to God solely on the basis of the imputed righteousness of Christ, do you mean that imputation alone, considered as distinct from sanctification, can actually make us fully acceptable to God so that because of it he no longer has any basis at all for any rejection of us but is in fact wholly pleased with us? Imputation, considered as distinct from sanctification, removes all barriers to God's finding us wholly acceptable? Or, rather, do you mean that imputation makes us fully acceptable to God in principle, but without bearing its fruit within us in sanctification it would remain in fact ineffective (because not fully actualized) and would thus not remove all the barriers to God's full acceptance of us?"
This is a hugely important question. If we go with the former answer, it would mean that God does not have any moral concerns about our inward state or our behavior. So long as Christ's righteousness is imputed to us, we could remain wholly unsanctified and yet be fully acceptable and morally pleasing to God. There would be nothing to keep us out of his presence or stop the full showering of his acceptance and pleasure coming down on us. Catholics cannot accept this point of view, for it trivializes sanctification and portrays God as morally blind. It turns salvation into a legal fiction, where God simply manages to fail to notice our sinful condition and treats us as if we are what we are not. Moral wickedness of character is something that is in its own nature displeasing to God, and we could not find full acceptance with God and be fully pleasing to him without its removal. This is evident to reason, and the Bible also speaks in this way. It testifies to us that we cannot be pleasing to God unless we are cleansed of our sins and made holy within. It tells us that God will judge us in the end not solely on the basis of some kind of imputation but on the basis of our works (done by the power of the grace of God). Clearly, to be rejected in the final judgment is not compatible with our being fully accepted by God!
However, if we go with the latter answer, Catholics can wholeheartedly accept the doctrine. Catholics agree that salvation is wholly of grace. As the Council of Orange (which Catholics fully accept) puts it (in Canon 22), "No man has anything of his own but untruth and sin. But if a man has any truth or righteousness, it is from that fountain for which we must thirst in this desert, so that we may be refreshed from it as by drops of water and not faint on the way." All saving good that we possess is wholly a gift of grace, merited for us by the suffering and righteousness of Christ. It is Christ's righteousness given to us as a free gift and not our own righteousness coming from ourselves which is the sole basis of our forgiveness of sins and our acceptance with God. As Canon 12 of the Council of Orange puts it, "God loves us for what we shall be by his gift, and not by our own deserving." Although Catholics do not use precisely the same language, they can fully accept the idea that it is Christ's righteousness imputed to us (credited to our account, given over to us as a free gift) which is the sole basis of our acceptance with God, but they would add that this gift is not complete unless what is imputed to us is also infused within us, becoming ours inwardly and impacting our actual life and behavior. We are not simply to be reckoned as righteous, but we are to become truly righteous through our union with Christ. If all we have is imputation, God would have given us with one hand what he takes away with the other, for we could never enjoy the benefits of imputation. It would be like buying a house but only ever getting the title deed and never actually getting to live in the house. This is why Catholics tend to include in the full idea of justification both the imputation component (that is, the giving over to us of what was before not ours) and the infusion component, as can be seen in the Catholic Catechism's description of justification above. Both are necessary to bring about our complete acceptance with God, not because Christ's righteousness is not enough but because there are two parts to how we receive it. But we don't want to get overly tied down with semantics. If Protestants want to use the term "justification" to refer only to imputation and "sanctification" to refer to the infusing and outworking of Christ's righteousness in our lives, this is not as important as what is meant by this language. So long as Protestants interpret their doctrine according to the latter way described above or in some similar way, so long as they admit that sanctification is needed to complete justification so that we would not in actuality attain full acceptance with God without it, Catholics can, I think, wholeheartedly embrace their point of view on this matter.
Surely this ought to count for something as we continue to dialogue with each other. We both look for salvation and acceptance with God wholly on the basis of the righteousness of Christ and not on the basis of our own personal merit. Isn't this the very core of the gospel doctrine of salvation?
For more on the Catholic doctrine of salvation, see here.
ADDENDUM 10/22/2015: A follow-up question for Protestants: "Do you believe that our good works merit God's favor?"
Let me clarify this a bit. Of course our works considered as ours do not gain us any personal merit before God, as if we could boast of them and say we had earned God's favor by means of them. All our goodness is an unmerited gift of God's grace. However, the inward holiness produced in us by the Holy Spirit in our sanctification and the good works that flow from that are truly morally good. That which is morally good "merits" God's favor. The word "merit" simply means that something warrants something else. The holiness given to us by the Holy Spirit, being good, is intrinsically pleasing to God, and so warrants his pleasure and favor. He cannot but be pleased with and thus find rewardable true holiness. Do you agree?
If you say that our holiness and our works do not merit God's favor in the sense defined above, this implies that our holiness isn't truly pleasing to God, that it does not warrant positive regard from him. Catholics cannot accept this idea, for both reason and Scripture testify that God is truly pleased with the good works of the just, and indeed will grant the reward of eternal life to our good works at the final judgment (what greater testimony of approbation could there be than that?). This does not give us any ground for boasting, for this goodness does not come from us and our own resources but from the grace of God through Christ. It is Christ's righteousness in us and not our own righteousness that we have produced for ourselves, and so all the glory goes to God for it. But Christ's righteousness in us surely is eminently good and pleasing to God. (Augustine puts the two parts together in this way--I'm paraphrasing--: "When God crowns our merits he is crowning his own gifts.") Therefore, a perfectly sanctified being with no remnant of sinfulness left in him could not be sent to hell by God, for hell is an expression of God's displeasure, whereas such a being would be fully pleasing to him and fit for heaven. God could not treat a sanctified being, whose character must be pleasing to him, the same as an unsanctified being, whose character is abhorrent to him--or he would be testifying that he has no regard for holiness and considers it no better than unholiness. It would be eminently unfitting to have a being who loves God with all his heart existing in hell for all eternity (for the same reason it would be unfitting to have a being who hates God with all his heart existing in heaven for all eternity).
But if, on the other hand, you say that our holiness and works, produced by grace in us, are indeed truly pleasing to God and truly warrant his pleasure and favor, and yet we cannot claim personal merit on their basis because they are gifts from God, Catholics fully agree with this. Again, as the Council of Orange puts it, "God loves us for what we shall be by his gift, and not by our own deserving." Catholics use the term "merit" in reference to the holiness God has produced in us not in order to claim that we deserve something from God of ourselves, but to make clear that what God works in us is pleasing and precious to him (since it reflects his own image) and thus warrants his favor and good will. If Protestants don't feel comfortable using the term "merit" in this way because they find it hard to separate the word from the Pelagian concept that we can earn God's favor by means of our own works produced from ourselves, that is not as important as the content of what they are trying to say. So long as Protestants can affirm that although our holiness does not give us a basis to claim to earn God's favor, it is indeed truly pleasing to him and thus warrants in its own nature God's pleasure and favor, we have no disagreement in terms of the substance of our affirmations on this point. And, again, I think this ought to count for something in our dialogue with each other.
ADDENDUM 6/3/16: Here is a sermon I wrote as a Protestant, but it reflects (in Protestant language and in a Protestant context) a view on the relationship between justification and sanctification that Catholics could in substance agree with. Here, here, and here you can find some criticisms of the form of the Protestant doctrine that has the problems I discussed above.
ADDENDUM 1/1/18: Here is another article I've just written up that gets at the same basic ideas and questions raised in this article in a bit of a different way. The article's main new contribution is that I've given labels to the two different interpretations of the Protestant doctrine of justification. I call the Catholic-friendly interpretation the Pro-Augustinian view and the Catholic-unfriendly interpretation the Anti-Augustinian view, in reference to how the two views compare to the substance of St. Augustine's ideas on justification.