Speaking from a stage in Parliament Square as part of Saturday's March Against Racism, Begg referenced a speech by the American president in which he said he would be sending more inmates to the controversial facility on the Cuban mainland.
He said: "The rise of the far right and the Nazis and fascists has seen a new wave with the election of Donald Trump, who said when he came to power, 'I'm going to load up Guantanamo with some bad dudes.'
"So my response is: 'When are you going, dude?"
Begg, from Birmingham, joined Labour MP for Tottenham David Lammy in slamming a small group of counter-protesters.
Suspected to be from the far-right political group the English Defence League (EDL), the group were spotted near crowds of anti-racism marchers on their way from Regent Street to the foot of Parliament.
He said: "We have smashed the EDL.
"They are no longer of any consequence. But let's not pat ourselves on the back too early, because some of their views have become mainstream."
Begg spent almost three years under US custody on suspicion of terrorist affiliations between 2002 and 2005.
He was later released and has spoken widely of his experiences as a prisoner in the Guantanamo Bay institution and other detainment camps under American control.
An estimated 30,000 people joined Saturday's march, which started outside the BBC's London offices in Portland Place at midday, before moving to Parliament Square at the base of the Palace of Westminster - now famous for scenes of political protest and upheaval.
Anti-racism supporters carrying signs with colourful slogans including "Migrants make our NHS" and "Black Lives Matter" led a peaceful, but spirited, course through the fashionable thoroughfares of Regent Street and Haymarket, banging drums and singing songs as they marched.
Mr Lammy said this week's meeting between former Ukip leader and key Brexit campaigner Nigel Farage and far-right French politician Marie Le Pen was evidence of current ill-feeling towards minorities.
He said: "This week Nigel Farage met Marie Le Pen and we are sending a message that we don't want that kind of fascism and xenophobia across our planet and in our country.
"My parents arrived in this county in the 1950s to signs that said 'no blacks, no Irish, no dogs' and we thought because of so much strife that by the time we got to the 21st century we had turned our backs on racism.
"And then, we get June the 21st. We get a rise in hate crime across the country almost by 50%."
Mr Lammy was not the only speaker to express concern at Mr Farage's recent appearance with Ms Le Pen ahead of the forthcoming French elections in April.
Fellow Labour MP Catherine West said the meeting had been the "worst image in all of the TV coverage this week".
She said: "This is a step towards an outcome that we are here today to say, 'no way', because Nigel Farage has always said he is a perfectly reasonable choice on the ballot paper.
"He is standing alongside a fascist - Front Nationale is a fascist organisation - and we must contact every single person we know in France to say 'you must not vote for them when it comes up on April 23'.
"We cannot have the Front Nationale in France. Regardless of any excuse that comes their way, the Front Nationale and Nigel Farage are of the same ilk."