Let's outline the actual case here.
Firstly, he wasn't fired because he wasn't employed by the organisations in question.
Hunt was an
emeritus professor. The very position means that he was retired, but retained an honorary position after that retirement, with such positions having varying levels of involvement in the actual running of the labs (often advisory while running side projects purely for the interest of the professor in question - ones that don't require funding grants - in my experience). He held such positions at a number of institutes and labs, of which he resigned three that I'm aware of; UCL, the Royal Society and the ERC.
The main position mentioned was at UCL.
According to their statement;
Sir Tim Hunt's personal decision to offer his resignation from his honorary position at UCL was a sad and unfortunate outcome of the comments he made in a speech last week. Media and online commentary played no part in UCL's decision to accept his resignation.
Sir Tim held an honorary position at UCL. He was not, and never has been, employed by UCL at any stage of his career and did not receive a salary from UCL.
UCL sought on more than one occasion to make contact with Sir Tim to discuss the situation, but his resignation was received before direct contact was established.
UCL accepted his resignation of his honorary position in good faith, and in doing so sent a clear signal that equality and diversity are truly valued at UCL. We continue to be open to engagement and dialogue on how we can best deliver on our commitment to these values.
The Royal Society and ERC are professional bodies belong to, but are not employers in this sense. He resigned a position on one of the Royal Societies boards, but retains his fellowship. I can't find any details of his position at the ERC at the moment.
He still retains a position at the
Crick Institute, which at least means he is somewhat involved in research in an emeritus fashion. Like most Nobel winners he likely made a considerable post-retirement income from touring and giving talks (controversy hasn't ever really blunted that, even in the case of the outright racist winners), while it's unlikely any of the honorary positions paid beyond use of facilities. So he is still in a research position and still has his primary source of income.
This isn't just semantics. This is organisations and an individual with a mutual affiliation deciding to end that affiliation, not an employer sacking an employee. This is an independently wealthy (from the Nobel and associated speaking circuit, on top of whatever pension and savings he has from a career as a senior researcher) retired man who lends his reputation to various institutions finding that reputation isn't worth as much any more, not someone losing their means of supporting themselves.
Secondly, this wasn't a single unsupported source resulting in an internet witch hunt. There were at least three initial scientists/journalists who broke the story.
Connie St Louis was the central source, but she was supported by Ivan Orlovsky (founder of
Retraction Watch and
Deborah Blum. All three are extremely experienced science journalists (as in, journalists with an actual background and understanding in science, not just journalists shoved onto the science beat repeating press releases), and Orlovsky holds an MD.
Blum had done a follow up informal questioning with Hunt at lunch the next day where he said he
stood by the core of what he said.
Thirdly, and linked to that last bit, the resignations came after an initial interview he gave with the BBC where he said much the same thing. He part-apologised but
stood by the comments. This is, to my understanding, the statement that lead to the resignations, and it does seem to admit fault.
As for this being a problem;
The more senior the head of a research lab, the less likely it is to employ women. This is a current problem, not historical. Women are still being quietly frozen out of the top levels of research, and it's hard not to see institutional views among the elite (like Hunt) as a contributory factor. The 'leaky pipeline' problem is widely recognised and a lot of effort is going into patching it. Casually sexist remarks going unchallenged (or being expected as the norm) is one of the common targeted issues.