Julian Lewis, chairman of the Defence Select Committee, said the former prime minister had "charm and ability".
But he had "dangerously" delayed renewing Trident, Mr Lewis said, also questioning Mr Cameron's roles in Libya, Brexit and armed forces cuts.
The ex-PM was mooted as Nato secretary-general over the Christmas period.
The incumbent, former Norwegian Prime Minister Jens Stoltenberg, is due to stand down within the next couple of years.
In a speech as part of the Speaker's Lecture series, Mr Lewis said: "During the Christmas recess, an obviously planted story appeared in a tabloid newspaper suggesting that our most recent former prime minister should be the UK's candidate as next Nato secretary-general.
"It was duly taken up by the wider media, with heavy hints suggesting that Number 10 was actively considering the proposal."
He added: "David Cameron is a man of charm and ability. He deserves to find a role commensurate with his talents, and I am sure that he will.
"But those talents do not include wisely judging strategic issues, whether when toppling Arab dictators in places like Libya, increasing military commitments whilst cutting the armed forces, predicting a Third World War in consequence of Brexit, or dangerously delaying the renewal of Trident for the sake of coalition politics - as he did."
Mr Cameron resigned as prime minister last summer after his Remain campaign lost the EU referendum. He stood down as MP for Witney, Oxfordshire, in September.
In an interview with the Times, published on Monday, US President-elect Donald Trump branded Nato "obsolete" for failing to adapt to the threat posed by terrorism, but he also said the 28-member military alliance was "very important".
Mr Lewis warned that if the US turned away from the organisation, Europe would have "little chance" of deterring non-nuclear threats.
But he claimed Mr Trump was trying to strengthen the alliance by confronting countries which failed to meet the membership requirement of spending more than 2% of GDP on defence.
Mr Lewis said: "It seems to me far more likely that, by confronting the 'free-riders' within the alliance, he actually aims to strengthen Nato by ensuring that all its members properly fund their armed forces in the future."
David Cameron should not become leader of Nato as his "talents do not include wisely-judging strategic issues", a senior Conservative MP has warned.