No, the European Parliament often rejects individual nominees for whatever reason. For example, Rocco Buttiglionne who was the Italian nominee I think to one of the Barroso Commissions was rejected for his views on homosexuals (which in my opinion should not have been a cause for rejection, but OK) by a majority in the EP. They threatened to veto the entire Commission if he was not taken out, and so he was replaced.

However, that is not the point. The members of the Commission are appointed by their own respective governments. So the British member of the Commission gets appointed by the UK government, which I believe is elected, no? Are the individual ministers in the UK cabinet also elected specifically or are they appointed by the PM? This story about unelected official is such a strawman.

Anyways ran across this comment like 3 years after it was published, but had to comment.