Nobody that I have seen said Eich shouldn't have any job anymore. Just that he wasn't a good choice for this particular leadership role.
I'm not privy to what happened within Mozilla, but it does sound like it was internal disagreements (i.e. Mozilla employees begin unhappy with the appointment) that lead to him leaving the post, more than bloggers or HN commenters. That is just the nature of taking a leadership role. If you can't take the staff with you then your position isn't tenable.
A CEO post have everything to do with the public perception you create of the company, though. And deservedly or not, his position coupled with his opinions on prop 8 have come to have an impact on the perception of Mozilla. In that respect, the personal views of a CEO often does have something to do with their personal views in a way that is rarely the case for less public-facing positions.
I'm not privy to what happened within Mozilla, but it does sound like it was internal disagreements (i.e. Mozilla employees begin unhappy with the appointment) that lead to him leaving the post, more than bloggers or HN commenters. That is just the nature of taking a leadership role. If you can't take the staff with you then your position isn't tenable.