Does Catholicism teach that non-Christians can be saved?

As a former RC (Roman Catholic), who was born again while still a Catholic and know from doctrine and experience, i would like to say that the real question is how many Catholics will go to Heaven.

The assertion that in Heaven all are Catholic is absurd. Believing on the risen Lord Jesus Christ by His sinless shed blood to saved you as a damned and destitute sinner is what saves, rather than partly trusting in a church or one's own merit. True faith effects obedience, including being part of a church as able, but the gospel of Catholicism is one that largely preaches itself, and fosters faith in its power and one's own goodness to save them.

As for what they believe, this is open to some interpretation, as is most of what Catholics believe and practice, despite claims to the contrary.

And by "Catholic" i presume that you mean Roman Catholics, for while they are in "communion" with the Orthodox (and other "Catholic"churches), they both claim to be the One True Church in particular, and differ in not less a substantial issue as papal supremacy and infallibility.

On this issue the historical position of Roman Catholicism has leaned strongly toward excluding all who were not in formal communion with her:

Pope Boniface VIII, Unam Sanctam:

“We declare, say, define, and pronounce that it is absolutely necessary for the salvation of every human creature to be subject to the Roman Pontiff.” — Pope Boniface VIII, Unam Sanctam (Promulgated November 18, 1302) http://www.fordham.edu/halsall/source/b8-unam.html

St. Thomas Aquinas: It is also shown that to be subject to the Roman Pontiff is necessary for salvation. St. Thomas Aquinas, Against the Errors of the Greeks, Pt. 2, ch. 36 http://dhspriory.org/thomas/ContraErrGraecorum.htm#b38

…that no one, whatever almsgiving he has practiced, even if he has shed blood for the name of Christ, can be saved, unless he has remained in the bosom and unity of the Catholic Church.” Pope Eugene IV, Cantate Domino, Bull promulgated on February 4, 1441 (Florentine style), proclaimed “ex cathedra” (infallible).

The Holy Catholic Church teaches that God cannot be adored except within her fold; she affirms that all those who are separated from her will not be saved. (Pope St. Gregory the Great, “Moralia,” XIV:5)

However, Rome has infallibly decreed that she is infallible (conditionally) and cannot change, and can define what she meant as needed.

Thus the RC Catechism (RCC: 846) states, How are we to understand this affirmation, often repeated by the Church Fathers?[335] Re-formulated positively, it means that all salvation comes from Christ the Head through the Church which is his Body:…Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it or to remain in it.[336]

847 This affirmation is not aimed at those who, through no fault of their own, do not know Christ and his Church:

Those who, through no fault of their own, do not know the Gospel of Christ or his Church, but who nevertheless seek God with a sincere heart, and, moved by grace, try in their actions to do his will as they know it through the dictates of their conscience - those too may achieve eternal salvation.337

The "through no fault of their own" clause is what is debated among Catholics, as Rome has not definitively clarified her ambiguity as to whether it excludes such souls as Protestants who known of the claims of Rome but "through no fault of their own" believe such Catholic claims are unwarranted, but these Protestants obey the light they have, or if it only refers to those who are ignorant of the claims of Rome, but obey the light they have.

In addition, even the second condition is open to interpretation as to whether it means that such will come to accept the Catholic church as a consequence of obeying the light they have, or whether those "through no fault of their own" who are ignorant of the Catholic church, or the gospel of salvation, can be saved.

The latter encounters a further problem in that Scripture indicates that outside souls who are not mature enough to make moral choices, (Is. 7:15,16) ignorance of truth is our fault, and that by obeying the truth we have then we will come to the Truth of salvation.(Mt. 13:11,12; Rm. 2:7; Acts 10)

In favor of Protestants being able to saved who know the claims of Rome but reject them in good conscience are the words of Vatican Two (which Traditional Catholics impugn or reject as being hijacked by liberals),

LUMEN GENTIUM: 16; The Church recognizes that in many ways she is linked with those who, being baptized, are honored with the name of Christian, though they do not profess the faith in its entirety or do not preserve unity of communion with the successor of Peter. (Cf. Gal. 4:6; Rom. 8:15-16 and 26)

For there are many who honor Sacred Scripture, taking it as a norm of belief and a pattern of life, and who show a sincere zeal. They lovingly believe in God the Father Almighty and in Christ, the Son of God and Saviour. (Cf. Jn. 16:13) They are consecrated by baptism, in which they are united with Christ. They also recognize and accept other sacraments within their own Churches or ecclesiastical communities.

They also share with us in prayer and other spiritual benefits. Likewise we can say that in some real way they are joined with us in the Holy Spirit, for to them too He gives His gifts and graces whereby He is operative among them with His sanctifying power. Some indeed He has strengthened to the extent of the shedding of their blood..

[Note: Rome does not refer to Protestant church as churches but "ecclesiastical communities," while she herself is guilty of shedding their blood because they obeyed in conscience toward God they obeyed light of Scripture.And the fact that Rome stipulate baptism evidences that it is not merely affirming salvation by obeying the limited light one has, but that she recognizes that Prots can be saved by believing the gospel and which saving faith is expressed in baptism, and which regeneration can take place before the person knows of or joins a church as seen in Acts 8, 10, etc.]

LUMEN GENTIUM continues,

But the plan of salvation also includes those who acknowledge the Creator. In the first place amongst these there are the Mohammedans, who, professing to hold the faith of Abraham, along with us adore the one and merciful God, who on the last day will judge mankind.

This portion in particular is troubling because it affirms Allah is the God of Abraham, which is blasphemous, as they are manifestly not the same. Nor is this the case of worshiping an unknown God as in Acts 17:23, but is a case of "identity theft" by Islam.

In addition is the inference that Muslims etc. can be saved apart from believing the gospel Christ in this life, due to ignorance thru no fault of their own and acting in good conscience, which many Prots wrestle with as well, (see Rm. 2) though it is clear that "Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved. " (Acts 4:12)