Absolutely not true. Stories like this have their origin in the first atomic bomb test, the Trinity test. Much earlier in the Manhattan Project, in 1942, Teller raised the question of whether an atomic bomb explosion might result in a thermonuclear (i.e., fusion) reaction in the atmosphere (nitrogen + nitrogen -> oxygen + carbon + energy). This was thought to be impossible, but the suggestion was checked, and the conclusion was that it was impossible. The only dissenting opinion was Arthur Compton, who was not convinced of the complete impossibility of it, but estimating the probability at less than 1 in a million.

In reality, the probability of such a thermonuclear reaction was zero. Truman authorised a test that had zero chance of blowing up the world (FDR died in May; the test was authorised after his death). Apart from Compton, the Manhattan Project scientists (correctly) believed that the probability of blowing up the world was zero (and even Compton though that the probability was much less than 50/50).

For a transcript of relevant parts of an interview with Hans Bethe on this, see https://blogs.scientificamerican.com/cross-check/bethe-teller-trinity-and-the-end-of-earth/

The idea that there was a significant chance of blowing up the Earth (and when it comes to blowing up the Earth, a 1 in a million chance can be considered significant) comes from a post-war interview of Compton by Pearl Buck, in which they discussed the possibility of a runaway atmospheric fusion reaction. Compton said that if the risk was greater than 3 in a million, the bomb should not be built, but the risk was calculated as a little less than that, so they went ahead.

Reference:

Pearl S. Buck, "The bomb - the end of the world?", The American Weekly, pg 8-9,119, 8th March, 1959.