Stephanie Callaghan is currently standing to be the MSP for Uddingston and Bellshill, replacing the retiring Richard Lyle, and is sixth on the party's regional list.
In 2017, Ms Callaghan tweeted about the decision of then prime minister Theresa May to refuse an independence referendum.
First reported in the Scottish Sun, the now deleted tweet said: "Tory propaganda provides a window into future plans: stamp on democracy.
"Hitler did same: set scene 4 Jewish Holocaust to lower opposition."
The Campaign Against Anti-Semitism said: "There is no comparison between political tensions in the UK today and Nazi Germany's systematic destruction of democracy and murder of six million Jewish men, women and children.
"Politicians must set an example by learning the lessons of the Holocaust - not diminishing the memory of those innocents who were slaughtered by using the Holocaust to score political points
"To make such a comparison is disgraceful and wounding, showing incredible ignorance."
Scottish Tory MSP Annie Wells said it is "beyond belief" that Ms Callaghan has not already been removed as a candidate.
She said: "It's absolutely appalling to compare stopping another referendum to Nazi propaganda and Hitler's actions.
"What kind of warped mind would think opposing independence is anything like the Holocaust?
"It's disgusting and completely out of order for someone standing for election.
"An apology won't cut it for such a vile remark.
"The SNP should have removed this candidate immediately.
"It's beyond belief that they seem content to keep a candidate who downplays the Holocaust."
Ms Callaghan told the newspaper: "The words in this old tweet were poorly chosen and I apologise for the offence caused.
"I have deleted the tweet.”