While this is an oldish question with some existing good answers, I think there's something fundamental missing from all of them: "Universal grammar" is both ambiguous and vague, and people who "don't believe in UG" are usually not even talking about the same thing as people who "believe in UG".
The core notion of UG is pretty uncontroversial: there is something specific to linguistic structure that's unique to humans, universal across humans, and indispensable to language as we know it.
The controversial part—the part people believe or don't believe—is what that "something" is.1 And it depends on who you ask—and when you ask them.
Since 2002, Noam Chomsky's position has been that UG is just recursion.2 Many people have argued against that, but the argument is usually that there's obviously more to language than just recursion, not less.3
People who argue against UG are instead arguing against Chomsky's early 90s position, which required UG to be a whole lot more heavy-duty, but which nobody believes anymore. Briefly, his paradigm at the time, Principles & Parameters/Government & Binding, strongly implied that UG included a large set of pretty complicated, language-independent principles like "no wh-movement outside an island."4
Since around 1993, GB has been replaced by the Minimalist Program,5 which implies that many of those principles don't have to be specifically built into some language organ; they follow automatically from the idea that language picks the most optimal path to get from sound to meaning.6 Exactly what ends up as still part of UG is hard to say, and doesn't really matter. If pressed, I suspect most people would say they don't know, but it's probably more than just recursion and much less than GB required.
In non-mainstream theoretical grammar approaches, almost everyone has learned that arguing from first principles gets them nowhere but ignored. They need to argue that their theory covers more of the data, or can be used to build an efficient parser (as demonstrated by writing the code for one), or can make psycholinguistic predictions that can be tested in the lab, etc.
So, who does care? Mainly people working in neighboring fields, like computational linguistics, neurolinguistics, or philosophy of mind. And to them, UG still means the version implied by the 1980s theory, or one of its predecessors.7 They sometimes even have pragmatic reasons for arguing against it.8
Are those people even "linguists"? If they are, does not believing in 1980s UG mean they don't believe in UG, or that they're defining it wrong? Who cares?
So, back to the question: Does the majority of linguists accept universal grammar? It depends on who counts as a "linguist"—but, even more, it depends on what exactly "universal grammar" includes. And no good answer to this question is particularly useful.9
1. There's also a lot of baggage attached to the idea that is controversial, such as the idea that UG is so different from every other mental module that it couldn't have evolved the same way. But let's ignore all of that baggage, because it's not actually part of UG.
2. The classic article is "The Faculty of Language". One of the authors, Hauser, has argued that UG is hooking up recursion to language. Chomsky himself has seemed to imply that UG is just recursion itself, and everything else we do with recursion is based on language, but it's not entirely clear. Also, Chomsky has gradually switched to talking about "the language facility, narrow" instead of "universal grammar". If you're interested in the debate on this, the back-and-forth exchange between HCF and Pinker and Jackendoff on most of the substantive issues is explained pretty well on Language Log, with links to the papers.
3. I should probably mention the whole Pirahã mess here. At first, both sides characterized that as a fierce debate about UG. But soon, the only thing the two sides agreed on is that recursion just means "Merge can combine things larger than words"—an individual language might restrict it so Merge can't combine an S inside another S or an NP inside another NP, but it's still recursive. Unless you believe there are languages with no sentences longer than three words, you're not arguing against LFN/UG. That doesn't mean the two sides stopped fighting, just that the fight wasn't about UG. And it leaves UG as something nobody can really argue against (except from the opposite direction—that there's more to language than recursion).
4. Most of the complexity is inside the technical definition of what an "island" is, so I don't want to explain it here. Just trust me, it's not trivial.
5. Explained in Noam Chomsky's 1995 book The Minimalist Program, but you're not going to understand that without a lot of background. Wikipedia's article isn't great, but maybe it works as a starting point.
6. Does that sound fishy? I'm not going to argue in favor of it. But it is certainly true that if you accept that the mind can feasibly do what MP requires, it doesn't need a heavy-duty UG anymore.
7. Basically, if you're building parser programs or modeling lexical storage in a neural net or whatever, Chomsky 1995 doesn't give you any useful tools that Chomsky 1986 or even Chomsky 1965 didn't. So there's really no reason to keep up to date. There are also some sociological factors not worth getting into here. Meanwhile, some of the alternative frameworks have gone out of their way to provide tools to the neighboring fields, but they don't expect you to first learn all of MP so you can unlearn it to pick up HPSG.
8. For example, you need to justify why you're not using the state of the art theory, or why you're using a minority theory instead of the mainstream one, having a good argument against P&P in terms of the UG it implies being impossible can get your manager/professor/committee/whatever off your back…
9. Of course in a few decades, when all of the relevant fields have caught up to the point where they can join up, then the question of "OK, exactly what should we consider part of UG (or 'Language Facility, Narrow', or whatever)" will be a good question. It may even have a good answer. But we're not there yet.