Are Protestants with a valid baptism Catholics?
The short answer is no, but that does not mean that they may not be saved nevertheless. Validly baptized Protestants may eventually be saved, but are not consequently Catholic.
Obviously a Protestant who does not accept all the dogmas of the Catholic Church, can not presume to be Catholic in any stretch of the imagination. For example, if a Protestant does not accept papal a uthority or some Marian dogma then he or she can not profess to be Catholic. Catholic dogma is for Catholics. Catholic sacraments are administered by Catholic clergy.
Many Protestant baptisms are recognized as valid by the Catholic Church, if done using the Trinitarian formula and using valid matter, ordinary water. Thus if the Catholic Church recognizes that such baptisms are valid, that means that sanctifying grace has entered the soul. That much is obvious.
Although Protestant baptisms are recognized as valid, if done in the proper manner, that also means that they remain Protestant and not Catholic, until that moment they make a formal act of faith in Catholicism.
The phrase extra Ecclesiam nulla salus has been around for a long time.
The Latin phrase extra Ecclesiam nulla salus (meaning "outside the Church [there is] no salvation" or "no salvation outside the Church") is a phrase referring to a Christian doctrine about who is to receive salvation.
The expression comes from the writings of Saint Cyprian of Carthage, a Christian bishop of the 3rd century. The phrase is an axiom often used as shorthand for the doctrine that the Church is necessary for salvation. It is a dogma in the Catholic Church and the Eastern Orthodox Church, in reference to their own communions. It is also held by many historic Protestant churches. However, Protestants, Catholics and the Eastern Orthodox each have a unique ecclesiological understanding of what constitutes 'the Church'. For some, the church is defined as "all those who will be saved", with no emphasis on the visible church. For others, the theological basis for this doctrine is founded on the beliefs that Jesus Christ personally established the one Church, and that the Church serves as the means by which the graces won by Christ are communicated to believers. - Extra Ecclesiam nulla salus
Modern Catholic understanding of the phrase Extra Ecclesiam nulla salus can be understood as follows:
Extra Ecclesiam nulla salus
Outside the Church there is no salvation. Solemnly defined doctrine that says that no one who culpably refuses to become and remain a member of the Catholic Church can be saved. Positively this means that all who reach their eternal destiny are saved through the Catholic Church, of which Christ is the visible head. This is true even though they nay not have lived as professed Roman Catholics (Fourth Lateran Council, A,D, 1215, Denzinger 802).
In the end, from a Catholic perspective a Protestant may be saved as long as they have kept their baptismal innocence intact before death, thus remaining in the state of grace. If having sinned grievously against God they should make a perfect act of contrition in order to restore their friendship with God. The same holds true for Catholics, but a Catholic who commits a mortal sin must go the confession and receive sacramental absolution or in case of urgency make a perfect act of contrition and then followed by a sacramental confession when possible.
The Vatican document Dominus Iesus explains the possibilities of being saved while not formally attached to the Catholic Church in which Christ is her head.
The 2000 declaration Dominus Iesus of the Congregation for the Doctrine of the Faith states that "it must be firmly believed that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is the mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and baptism (cf. Mk 16:16; Jn 3:5), and thereby affirmed at the same time the necessity of the Church which men enter through baptism as through a door." It then adds that "for those who are not formally and visibly members of the Church, salvation in Christ is accessible by virtue of a grace which, while having a mysterious relationship to the Church, does not make them formally part of the Church, but enlightens them in a way which is accommodated to their spiritual and material situation. This grace comes from Christ; it is ... communicated by the Holy Spirit; it has a relationship with the Church, which, according to the plan of the Father, has her origin in the mission of the Son and the Holy Spirit."
In the end, Protestants may certainly be saved, their eternal destiny are saved through the Catholic Church, of which Christ is the visible head.
On the last day, there will be many surprises on both sides of this fence. We will be very surprised to see some of those who are saved and in heaven; and some of those who are condemned to eternal damnation. God knows what man is and knows what he is capable of, for good or evil. Oh that terrible day!