The question "is Catholicism Christian?" is a lot more controversial than it needs to be. Some people are offended by the very existence of the question, and will be quite angry if you give them an answer they don't like. Others will quite happily answer "no", and are so aggressive about it that they'll actually bash those who merely claim to be Catholic. Some Christians, either not knowing the teachings of Catholicism or not wanting to offend their erroneous Catholic friends, will either sit on the fence, or softly tread on one side or the other. But the simple answer to the question "is Catholicism Christian?" is no.
Now, this does not mean that anyone associated with the Catholic Church is necessarily unChristian. In fact, a good friend of mine was saved thanks to a Catholic priest who effectively works as a sheep in wolf's clothing. He does not teach his congregation the errors of his Church, but rather, he feeds them the word of God as any good pastor would do. On top of this, many Catholics do not know the teachings of their Church, and will even vehemently deny those teachings when pressed. You can cite Council rulings, Catechism of the Catholic Church, or even a few Popes, and they will reject the teaching.
Furthermore, while the Catholic Church does have some unbelievably bad doctrines, it does still have enough of the Christian faith that a Catholic may well be saved by their Church. I believe I have met many Catholics who are, in fact, saved, though I obviously cannot say for sure, nor can I deny that much of what they believe is an affront to God.
The simplest way to determine whether Catholicism is Christian or not is to figure out how one is saved in Christianity and see if Catholicism teaches the same thing. How do we do this? Of course, we go to scripture.
First and foremost, according to Christianity, we should "...remain faithful to what you have learned and believed, because you know from whom you learned it, and that from infancy you have known [the] sacred scriptures, which are capable of giving you wisdom for salvation through faith in Christ Jesus. All scripture is inspired by God and is useful for teaching, for refutation, for correction, and for training in righteousness, so that one who belongs to God may be competent, equipped for every good work." (2 Timothy 3:14-17). All Catholics should affirm this (and yet, ironically, cannot), first of all because I used a popular Catholic translation. While I would normally cite the more popular KJV, or at least one of my preferred study translations, I like to use Catholic translations when speaking to Catholics. But also, Catholicism officially teaches that "...both Sacred Tradition and Sacred Scripture are to be accepted and venerated with the same sense of loyalty and reverence".
Therefore, we actually have common ground. Both Christians and Catholics acknowledge that scripture is inspired by God and is authoritative. In fact, while it is impossible in practice, in theory, Catholics should treat scripture "with the same sense of loyalty and reverence" as Catholic tradition. In other words, they can't deny scripture without also denying Catholic tradition.
The trouble is, they also cannot accept scripture without denying Catholic tradition, at least when it comes to something so critical as salvation. According to scripture, "For by grace you have been saved through faith, and this is not from you; it is the gift of God; it is not from works, so no one may boast. For we are his handiwork, created in Christ Jesus for the good works that God has prepared in advance, that we should live in them." (Ephesians 2:8-10). Yet, according to the Council of Trent, a Catholic Council called as a response to the Reformation, "If anyone shall say that by faith alone the sinner is justified, so as to understand that nothing else is required to cooperate in the attainment of the grace of justification, and that it is in no way necessary that he be prepared and disposed by the action of his own will: let him be anathema".
Hey Paul, I know you're an Apostle, hand chosen by the Living Christ, whom you saw with your own two eyes, and that you are less than the most eminent Apostle in nothing, but uh... yeah, you're anathema. How can this be considered Christian? I haven't even gotten started. All I did was compare one extract of scripture to one canon from a Catholic Council and already found irreconcilable differences, to the extent that the Catholic Church declares Paul, an Apostle of God, anathema, for writing what he was inspired to write by the Living God!
According to the Christian faith, salvation is for those who confess Jesus as Lord and believe in their hearts God raised Him from the dead (Romans 10:9). The one who does not work, but believes in God who justifies the ungodly, will have his faith credited to him as righteousness (Romans 4:5). If we are saved by grace, as Ephesians 2:8-10 clearly says, then we are not saved by works, otherwise grace is no longer grace (Romans 11:6). Need I go on? It is the consistent testimony of scripture that salvation is by grace (the unmerited, and indeed demerited favor of God) through faith, as the gift (i.e. thing that need only be received) of God, and not of works (as in you literally don't need to do anything).
Yet, Catholicism teaches a works based system of salvation. This is a major difference. Christianity teaches salvation by grace, through faith, not of works, Catholicism teaches salvation by grace, earned through faith and works. The two are literally irreconcilable. The false gospel of Catholicism cannot be reconciled with the true Gospel of Christianity. Therefore, as hard as many people will find it to hear, Catholicism is not a Christian religion.
If you're a Catholic reading this, you may well be offended. Of course, so would a non-Christian Jew upon being told their Messiah was both crucified and rejected by the majority of his people, so my first encouragement to you would be not to let your emotions cloud your judgement. Rather, first of all, consider what I have said. Compare the teachings of scripture on salvation to the teachings of your Church. But I would also encourage you to look at the examples of salvation in scripture and compare them to yourself.
In my opinion, by far the greatest of these is the penitent thief. The penitent thief was indisputably saved by faith alone because faith alone was all he had. Faith alone may not be all you have, but it is all you have that Jesus wants. Come to Him as did the penitent thief, then go from Him as did the disciples: Walk in the good works that God prepared in advance.