One argument Catholics love to throw around is the alleged unity of the Catholic Church compared to the alleged chaos of the large number of "Protestant" denominations (usually anywhere between 30,000 to 60,000). To even begin to respond to this, we must first ask, what is a Protestant denomination?
A Protestant denomination, by definition, is any nominally Christian Church that rejects the authority of Roman Catholicism. Some definitions add "in accordance with the principles of the Reformation". However, adding this qualification significantly reduces the number of "Protestant" denominations Catholics can appeal to. As shown here, at least some Catholics who make this argument appeal to denominations like Mormonism (and others) as Protestants, even though these do not hold to the principles of the Reformation.
Now, already we have a problem here. The distinction between Catholics and Protestants is essentially "us" and "everyone who is not us". "Catholic" is a "positive" term, designed to encompass a number of beliefs held by those who proclaim the name. "Protestant", by contrast, is a "negative" term, designed to encompass only one thing: "These people don't agree with us". So, already the playing field is tilted in the Catholic Church's favour. The irony is that literally any denomination can do this. As I like to say, if there are 40,000 Protestant denominations, that would just make Catholicism number 40,001.
Personally, I have no denominational affiliations, but for sake of argument, let's just pretend I'm an Evangelical. I will now make up a word for non-Evangelicals: Protesians. Well, now the Catholic Church is one of 40,000 Protesian denominations. That's proof that they're all wrong, and Evangelicals are right! How bad an argument would that be? It would be abysmal logic. It would be an abysmal argument if I made it, it's an abysmal argument when atheists use it to defend atheism, it is an abysmal argument when Catholics use it to defend Catholicism.
Now, let's go back to the Protestant denominations. First, how many are there? In truth, not as many as Catholics claim. The number is artificially inflated in two major ways. The first, as I've already alluded to, is to define "Protestant" as loosely as possible. If we allow for the addition of "in accordance with the principles of the Reformation" to the definition, already most of the division disappears. Mormons, as I said, do not hold to the principles of the Reformation, and ironically have more in common with Catholics than with, for example, Baptists. Likewise, Jehovah's Witnesses have more in common with Catholics than Lutherans. We could go through many "Christian" denominations that are opposed to Reformation principles, ironically including the Orthodox Church, which Catholics often refer to as "the original Protestants". Evidently, the "original Protestants" do not reject the Catholic Church in accordance with the principles of the Reformation, and these definitely have more in common with the Catholic Church than any Reformation-based denomination.
So that's one way Catholics inflate the number of "Protestant" denominations. The second is to take individual beliefs and make them out to be a whole new denomination. Calvinism, for example, is not a denomination (and ironically is just a re-vamped version of some of Augustine's beliefs), but rather, a specific way of looking at the nature of salvation as it pertains to predestination. Calvinism spans across many denominations, and is believed or rejected by a range of people within them all.
Furthermore (and here's where we get to the bones of the argument), it is absolutely trivial. While some divide over Calvinism, the truth is Calvinism itself does not generally affect the Church. Calvinists and non-Calvinists are perfectly capable of uniting in spite of our differences. This is the case with all of the ~1,000 Bible-based denominations that exist. All differences between us are trivial and not worth fighting over. One Church baptises infants, the other does not, both are considered Christian. One Church has an hour long sermon on a Sunday morning, another only gathers on Saturdays. Both are considered Christians. One Church foregoes the use of instruments in a service, another has a highly professional sound system. Both are considered Christians. The exclusive nature of the Catholic Church, which historically has taught that there is no salvation except through them (though of course, they have evolved since then) is unique to heretical denominations.
But one thing that is unfortunately not unique to heretical denominations is division from the inside. Humans are human. We sin, we divide. No denomination is free from "rogues". Not. Even. Catholicism. In fact, as Catholicism is the largest denomination, it actually has the largest number of rogues. Catholics are so divided against each other that I actually see more unity between two "Protestant" denominations than between two Catholics. I've had to switch tactics in almost every debate I have ever had with a Catholic, for they are as divided against each other as all denominations combined. Some Catholics swear undying loyalty to the Pope, some wish the current Pope would retire (or even die). Some believe the Catechism of the Catholic Church is authoritative, others believe it is worthless. Some bash the Bible, others respect it so much, they will dust it off and kiss it before putting it back on the shelf. Some believe abortion is a mortal sin, others believe it is a human right. Last time I checked, contraception, which is famously condemned by the Catholic Church (as per Encyclical Letter Humanae Vitae of the Supereme Pontiff, 1968), is approved, or at least not opposed by 89% of Catholics. The numbers may have changed since then, and I can't be bothered to verify (do your own fact checking), but safe to say there were actually more Catholics united against the Catholic Church than within it.
So how does this denomination argument stack up? In truth, it stacks against the Catholic Church. It stacks against the Catholic Church because it shows the fallacy of opposing "Protestants". It stacks against the Catholic Church because, in all honesty, it backfires on account of divisions within their Church. It stacks against the Catholic Church because it shows that in spite of not being bound by human authority or tradition, "Protestant" denominations still manage to unite on more important issues, even with our tiny issues remaining in conflict. My friends, this argument is an embarrassment to the Catholic Church, and yet, if we were to compile a list of the most common arguments used by Catholicism, this argument would feature prominently near the top. So what does this say about Catholicism as a whole?
The irony here is that the division among Catholics means that there are many ways "Protestants" can unite with many of them. Some Catholics are so Protestant, they are Catholic in name only. Many a believer resides within the Catholic Church, whether through ignorance of its teachings, or just a rejection of them. Nevertheless, the Catholic Church remains a false, heretical organisation. Christians ought to leave it, and non-Christians ought to repent of it. Let us ensure that all divisions are between God's people and Satan's lies.