So did NPR do the right thing?
First, as to Williams' First Amendment rights, a citizen generally only has free speech rights in regard to government action. If a private employer wants to fire someone for a something offensive they say they generally can. If the government fires someone for their speech there is a much higher standard involved. While NPR is called National "Public" Radio, the reality is that only a small amount of their funding comes from government sources. About 16% of NPR's funding comes from local, state, and federal government sources. Much of that funding comes from government grants which NPR applies for as part of a competitive process. Just like a private company which applies for a tax credit, or for a government grant, NPR does not lose the rights to hire and fire people like a normal employers simply because they receive some government funding. Conservatives are already calling for the federal government to "defund" NPR, thinking this will shut down the media operation. In reality, if Republicans succeed in defunding NPR it will hurt the organization for sure, but unlikely shut the broadcasting network down. In fact, liberals may rush to the defense of NPR with more private donations if conservatives succeed.
Simply because NPR had the right to fire Williams, it does not mean that necessarily should have fired Williams. In their statement on the firing NPR stated, "His remarks on The O'Reilly Factor this past Monday were inconsistent with our editorial standards and practices, and undermined his credibility as a news analyst with NPR." NPR does have a reputation among liberals as a more enlightened, tolerant network. Liberals generally also believe (while I know conservatives disagree) that NPR sticks to the facts in their broadcasting, avoiding much of the slanted coverage seen on CNN, MSNBC, and Fox News. Liberals may be wrong in this belief, but the reality is that much of NPR's audience is built upon this liberal base who believes in the network's objectivity. NPR was likely correct in believing many of their viewers would be bothered by Williams' statements. In the past Fox News has refused to extend contract of some liberal voices who do not agree with their overall conservative narrative. In the same way, NPR is making the decision that Williams' comments took him too far away from their brand. Only time will tell whether NPR made the right business decision.
As to the political correctness issue, ultimately "politically correct" is a very vague term, if one believes it exists at all. If there is a standard for political correctness, what makes one statement correct and another statement incorrect? If the standard is what the majority of people find to be offensive then Williams' comments were probably not "incorrect." If the standard is what comments would be offensive to a fairly large minority (Muslims), then Williams' comments likely were "incorrect."