Mr Kennedy told the Prime Minister he could have exerted influence in private over the nomination of Jean-Claude Juncker for a top Brussels job.
But the Conservative leader shot himself in the foot by pulling the Tories out of the largest group of parties in the European Union, Mr Kennedy said.
His attack came as Mr Cameron attempted to place the blame for the humiliating defeat last Friday, when he was outvoted by other European leaders by 26 to 2, squarely at Labour's feet.
The Tory leader accused the main opposition party of setting the country up to fail, by giving up the UK's EU veto.
Earlier, Mr Cameron had been jeered by Labour MPs who shouted "26-2" as he stood to report back on last week's European Council.
For his part Labour leader Ed Miliband described the result as an "utter humiliation" and accused Mr Cameron of a "master class" in how to alienate your political allies.
Mr Cameron has described Mr Juncker's nomination as a "bad day for Europe", and warned it means the UK is now closer to the EU exit door. But he has been hailed by the SNP who say they can "do business" with the pro-federalist.
Mr Cameron had a telephone call with Mr Juncker at the weekend in which he congratulated him on his nomination.