Nicola Sturgeon released a statement congratulating Mr Trump after his surprise victory in the US election.
But speaking at First Minister’s Questions the following day, she made clear that she stood by her previous criticisms of his campaign.
Ms Sturgeon had backed Hillary Clinton ahead of the presidential election.
The first minister told MSPs she had faced criticism for breaking the convention that political leaders do not comment on election campaigns in other countries.
But she said many of Mr Trump’s comments during the campaign had been “deeply abhorrent”.
She said: “I never want to be, I am not prepared to be, a politician that maintains a diplomatic silence in the face of attitudes of racism, sexism, misogyny or intolerance of any kind.
“I hope that president-elect Trump turns out to be a president very different to the kind of candidate he was and reaches out to those who felt vilified by his campaign.
“But people of progressive opinion the world over, I think, do have to stand up for the values of tolerance and respect for diversity and difference.”
Ms Sturgeon added that she wanted to “engage positively and constructively with the American administration”, but would “never shy away from standing up for these important principles”.
The first minister also said the Scottish government fully assess the potential impact for Scotland once the new president forms an administration and its priorities are made clear.
Scottish Labour leader Kezia Dugdale said Mr Trump’s behaviour towards women had sent a “dangerous signal across the world”.
She added: “In January we will have a misogynist in the White House who has boasted about assaulting women and has used the most degrading language possible.
“And, of course, Donald Trump’s tolerance is not just aimed at women – we all remember the sickening sight of him mocking a disabled journalist. We can’t forget his plans to build a wall or ban people of one faith from entering America.”
Mr Trump, whose mother was Scottish, has had an uneasy relationship with the country in recent years.
Ms Sturgeon stripped him of his role as a business ambassador for the Scottish government in the wake of his call for Muslims not to be allowed into the US.
Mr Trump also had a high-profile war of words with Ms Sturgeon’s predecessor, Alex Salmond, over the Scottish government’s backing for a windfarm near the tycoon’s Aberdeenshire golf course.