The awkward encounter saw each grip the other’s hand so firmly that their knuckles turned white, according to the BBC.
Mr Macron told French media he had wanted to “show he would not make small concessions, not even symbolic ones, but also not overdo things”.
The leaders met in Brussels on Thursday ahead of a Nato summit.
As their vigorous handshake continued over several seconds, Mr Macron and Mr Trump also looked each other fixedly in the eyes until the US president attempted to disengage.
Mr Macron told the Journal du Dimanche newspaper his approach to the encounter had been about getting respect.
“Donald Trump, the Turkish president or the Russian president see things in terms of power relationships, which doesn’t bother me,” he said.
“I don’t believe in diplomacy through public criticism but in my bilateral dialogues I don’t let anything pass. That is how you get respect.”
Mr Trump’s hand contact with foreign leaders has been closely scrutinised since he took power.
- In January he held UK Prime Minister Theresa May’s hand as they walked together – Mrs May later said he was “being a gentleman”
- He then yanked Japanese Prime Minister Shinzo Abe’s hand towards him in one of his signature moves
- But Canadian Prime Minister Justin Trudeau prevented this by gripping Mr Trump’s shoulder during their encounter
- And in March Mr Trump appeared simply to ignore the offer of a handshakefrom German Chancellor Angela Merkel