The approach may have garnered praise among the sizeable number of euroskeptics in the U.K., but it has rankled a large part of the 28-strong EU.
Alexander Stubb, who took over as Prime Minister of Finland on Tuesday, told CNBC: "In the UK some people really seriously need to wake up and smell the coffee. The EU is a very good thing for the United Kingdom. Over 50 percent of the trade of the UK goes to the EU... if that is to be cut off, I think the continent will be cut off seriously."
However, he added that he believes the situation is salvageable.
"David Cameron is a very principled man and he has felt strongly that Juncker is a little bit too federalist for his liking. But I think we can all make amends. I think we can all solve the situation," Stubb said.
Polish Foreign Minister Radek Sakorski expressed his views more succinctly in a taped private conversation which was leaked to the media earlier this week when he said Cameron had "f***ed it up."
As well as objecting to Juncker personally, Cameron also took aim at the process which looks set to bring the former head of the Eurogroup in. Rather than being elected by direct democratic vote, Juncker will get in because he has the support of the European People's Party, the grouping of moderate right-wing EU parties which is the largest grouping in the European Parliament. Cameron's Conservative Party left the group in 2009.
Cameron runs the risk of appearing to simply pander to anti-EU voices within his own party, and to those voters who may be lured by the anti-EU UK Independence Party in next year's elections.
No-one cares that much about who's going to be the President of the European Commission, it's a British-only obsession," according to Gilles Moec, chief European economist at Deutsche Bank.
"The problem really in the euro zone is that when you look at the sources of growth, most of it comes from domestic demand, and that's almost a mechanical reaction to the fact that fiscal austerity is a bit less painful."
Cameron has pledged to re-negotiate the U.K.'s relationship with the EU, but by neglecting the back-room diplomacy which usually accompanies such appointments, he may in fact have scuppered his chances of winning more concessions for the country from other EU leaders.
Some were prepared to hold out an olive branch, although Hungary is believed to be the only country which will vote with the U.K. against Juncker's appointment.
Fredrik Reinfeldt, Prime Minister of Sweden, told CNBC he shared some of Cameron's "complaints" about the process by which Juncker is likely to be elected.