Marsch made the comments on Thursday, ahead of Canada's opening World Cup fixture against Bosnia and Herzegovina, while praising the commitment shown by his current squad.
"In the US, sometimes we had to beg players to sing the national anthem," Marsch said, before contrasting that with his Canada squad, who he said "belt it out to the top of their lungs" out of pride in representing their country.
Marsch served as an assistant to USMNT head coach Bob Bradley at the 2010 World Cup in South Africa, a tournament in which Dempsey was a key member of the squad that reached the round of 16.
Former Fulham and Tottenham striker Dempsey, now working as a Fox Sports analyst at this tournament, was asked about Marsch's remarks on Friday, ahead of Canada's match against Bosnia and Herzegovina.
"He really said that? Man, I can't take this guy too seriously," Dempsey said, rejecting the suggestion that he or his teammates lacked pride in representing the United States.
The Texan, the USMNT's joint all-time leading scorer alongside Landon Donovan with 57 goals, explained that his own pre-match routine simply did not involve singing.
"It was an honour for me growing up to represent my country," Dempsey said. "When the national anthem happened, I wasn't someone who normally would sing. I put my hand over my heart, and I'd pray to the good man upstairs."
He went further, pointing to the physical toll his international career had taken as evidence of his commitment to the United States.
"I'm someone who's bled for this country. I broke my nose playing for this country. I've come back from two heart procedures and played for this country," Dempsey said.
Dempsey then turned his attention to Marsch's own position, questioning the suitability of an American-born coach making such comparisons while leading a rival nation.
"I'm not going to take advice from someone who switched to the other side and is singing another country's national anthem," Dempsey said, before referencing former teammate Thierry Henry's catchphrase to close out his response.
"And as my boy would say, stay in your own lane. It looks like he's in a dang moped, so worry about your own team," Dempsey added.
Marsch's Canada side earned a 1-1 draw with Bosnia and Herzegovina on Friday, the first point in the country's World Cup history, with goals coming from Jovo Lukic and Cyle Larin.
For Marsch, the result offered some respite from the row, though the exchange with Dempsey is likely to follow him through the remainder of Canada's tournament campaign.