Alan Nimmo posted a cartoon on his Facebook page showing Mr Salmond and the Iranian chief, Mahmoud Ahmadinejad, side by side.
The caption read: "One of these men is a megalomaniac intent on increasing his dictatorial powers ... and a strong determination to destroy Britain. The other is the president of Iran".
The education convener of Labour/Conservative-controlled Falkirk Council later removed the cartoon, but Nationalists said he should stand down.
Leader of the SNP opposition on Falkirk Council Mrs Cecil Meiklejohn said Mr Nimmo's "use of inappropriate language" was "worrying".