Red Bull driver Verstappen was subject to booing and jeering from the crowd during his post-qualifying interview in parc ferme on Saturday, having qualified second for the British Grand Prix.
There were also cheers from the crowd when Verstappen spun exiting Stowe at the start of Q3.
Verstappen brushed off the boos when speaking in the FIA press conference later on, saying it was "a bit disappointing" that he could not hear the interviewer, Billy Monger.
"If they want to boo, they do that," Verstappen said. "For me, it's not going to change anything. I'm happy to always be here. It's a great track, great atmosphere in general.
"Maybe some of them don't like me, but that's fine. They all have their own opinions. I don't care."
Verstappen defeated Hamilton in a fierce fight for the championship last year in Abu Dhabi, winning the title in controversial fashion following the erroneous safety car restart ahead of the final lap.
But Hamilton made clear he did not agree with the booing at his home grand prix, saying: "I think we are better than that.
"I would say we don't need to do booing, but we have got such great fans and our sporting fans, they feel emotions, ups and downs, but I definitely don't agree with booing.
"I don't think we need to do that. It doesn't make any difference when you boo someone either, they have already made the mistakes or whatever it is.
"But I really do appreciate the support I have had here. I don't know, maybe some of them are still feeling the pain from last year still. Either way, I don't agree."
It is not the first time there has been booing amid the rivalry between Hamilton and Verstappen. At last year's Hungarian Grand Prix, in the wake of their crash at Silverstone, Hamilton was subject to boos from the crowds.
A number of drivers have discussed abuse they have received from fans in recent weeks. On Thursday, George Russell revealed he had been booed by a random fan in Montreal, and called for abuse to be "stamped out".
"We shouldn't be seeing any booing in any sports," Mercedes F1 boss Toto Wolff said after qualifying.
"I think that's unsportsmanlike. It's clear that we love the support the drivers have here and the team, that's fantastic, and the enthusiasm. But if you're not into the other guys, just remain silent. That would be a good way.
"The booing, I think none of the drivers deserve any booing whatever happened last year, whatever the competition is."
Although Wolff said it was good that fans were "emotionally engaged" with F1, he said the booing was "a step further".
"Imagine yourself standing out there giving an interview or being on the podium, being booed is abusive," Wolff said.
"That's why the emotional sport that is controversial. [If] you are a fan or not a fan of one or the other driver, that's good. But there is a certain limit that we shouldn't overstep."
