I don't mean leaving the EU. That is most definitely a wise and smart thing for the UK to do. It is very much in the UK's interest, before the whole shoddy edifice completely self-destructs.
HOWEVER, it is very, very much in America's interest for the UK to stay in. Although it has relatively declined Europe is still a very important player on the world stage and contains some of America's major trading partners and military allies. Having an English speaking nation in the councils of Europe, and in particular one that is very close to the USA in its own right, is perceived by the US State Dept as being like having a free seat at the EU table. It certainly makes it easier for the US to relate to the rest of Europe (and vice-versa, incidentally). So in that sense, even though it comes over as being pretty high-handed, Obama is very definitely acting in America's best interest by putting pressure on the Brits to stay in.
This is a major problem for Britons like me who want the UK out of the EU. It's going to be a tough enough fight to win the argument with our own electorate without having to deflect interference from the US government, who we can't influence at all. If they want to, they can put a lot of pressure on our social, political and business institutions.
And if we DO manage to leave, it's bound to affect Anglo-American relations pretty badly. If nothing else Britain will simply not be as important to the USA as it now is. I don't quite know how we square that circle without ending a beautiful friendship.