Glenrothes MP Peter Grant made the remark while replying to a post from veteran broadcaster Andrew Neil.
It came after the GB News founder shared a Tweet from the Auschwitz Memorial that highlighted a Jewish toddler from Hungary who was murdered by the Nazis before his first birthday.
He wrote: "As accusations of fascism are bandied about today like confetti by the ignorant, ludicrously devaluing the word of any meaning, a reminder of what real fascism can do. And of its unconscionable evil."
In a now deleted tweet, Mr Grant replied: "You're more right than you care to admit. Murdering babies wasn't on the Nazi manifesto.
"Not until they'd been in power several years & stoked up fear & hatred against innocent citizens.
"Then, and only then, did they show their true colours."
Now Board of Deputies of British Jews president Marie van der Zyl has urged MPs to be careful with their language.
She said: “We are disturbed by the suggestion from some MPs that Nazism only gradually revealed its true aims.
“In reality, Hitler was always open about his aims – in 1925, well before the Nazis came to power, he had already written in Mein Kampf about the need to 'exterminate … the international poisoners of our people' and how thousands of Jews should have been 'subjected to poison gas'.
"The overwhelming majority of comparisons to the Nazis are extremely inappropriate, and we would urge people, particularly Parliamentarians, to choose their words with far more care.”
Scottish Conservative chief whip Stephen Kerr claimed the Tweet was “beyond the pale”.
He said: “For an elected SNP MP to post this was hugely offensive as well as being completely inaccurate.
“It beggars belief that any elected representative would think this sort of language was appropriate as part of a political debate. It has absolutely no place in civil discourse.
“This was a warped tweet and gave a worrying insight into what this SNP MP believes. Peter Grant must urgently apologise and reflect on this shameful behaviour.”
Lib Dem MP Alistair Carmichael said: "The horrors of what the Nazis did are of a level that comparisons are rarely acceptable.
"Peter Grant's comments and implied comparison are nowhere near acceptable or appropriate. If he is to remain in the SNP then he should withdraw his comments, apologise and undertake to get some proper education on the Holocaust."
Mr Grant has now apologised, saying: “I want to apologise unreservedly for a highly insensitive tweet I posted.
"While I strongly believe we must always be vigilant to the seeds of racism, anti-Semitism and fascism, I deeply regret how I made that point and I have deleted the tweet.”
Mr Grant is a former leader of Fife Council and was previously the SNP's Brexit spokesman at Westminster.
In 2018 the SNP apologised after posting a tweet appearing to mock the Tory minister Jacob Rees-Mogg, a Catholic, over his religious beliefs.
The MP had posted: “Jacob is a Catholic Brexiteer. He Fridays he only sells out fishermen.”