The specific group of organisations he mentioned was 'organisations with acronyms'. That would include almost every large professional organisation, so it's not really a very specific group of organisations.
Watching the interview continue, Gove makes no attempt to qualify his remarks by saying anything along the lines that he's only talking about a few people. He never acknowledges that most experts in most fields do actually know what they are talking about - which you would reasonably expect him to do if he didn't mean to rubbish all experts. And to my knowledge, he's never since then saiid anything to qualify his remarks, or anything in defence of 'experts' in general. Instead he talks about 'a faith in the British people to make the right decision', which in the context looks to me very much like an attempt to pitch the 'British people' against 'experts'. He then alleges that the experts have a vested interest in our continued EU membership (in other words, he's using the same tactic that - for example - climate change deniers use when presented with scientific evidence: Dismiss it by making out that the experts presenting the evidence are only doing so because they have a vested interest).
Realistically, putting those remarks into context, I cannot see any way that a reasonable person could fail to conclude that Gove was trying to trash the concept of listening to expert, professional, opinion. So saying (implied: we've) 'had enough of experts' by itself does amount to a slight misquote in terms of the actual words used, but seems to me to perfectly capture Gove's intended meaning.