Washington (CNN) Democrats in the US House of Representatives are widely rejecting a recent rallying cry by one of their colleagues to harass and push back against members of President Donald Trump's Cabinet in public spaces.

While Democratic leaders have already been clear in their criticism of the comments by California's Rep. Maxine Waters, rank-and-file members are following suit, saying her approach puts Democrats in an unsettling position, as members of her party worry about ceding the moral high ground -- which they consider their turf -- to Republicans.

"I think she's wrong on this," said Rep. Dan Kildee of Michigan. "We don't win by playing Donald Trump's game, by Donald Trump's rules."

Waters over the weekend encouraged the practice of heckling and harassing members of Trump's Cabinet as a way to vent frustration about the administration's recent "zero-tolerance" policy on immigration, which has led family separation at the border.

"Let's make sure we show up wherever we have to show up. And if you see anybody from that Cabinet in a restaurant, in a department store, at a gasoline station, you get out and you create a crowd," she said. "And you push back on them. And you tell them they're not welcome anymore, anywhere. We've got to get the children connected to their parents."

While Kildee, said he understood where Waters was coming from, he argued that such an aggressive approach could backfire.

"To invade people's personal space and their personal lives is something we wouldn't want to invite upon ourselves," he said. "Perhaps it's with a stronger sense of morality, but it's still defaulting to the tactics of the other side, and I think it's dangerous."

Rep. Steve Cohen, a Tennessee Democrat, argued that people are "just fed up" and they've reached a point where they feel like they have no other options to be heard. Still, he added, people deserve to live their private lives.

"It's tough being a public official these days, and I think you ought to be able to have some private time and not be set upon if you go to a ballgame, or if you go to a play or go to dinner," Cohen said.

Their comments represent a delicate fight Democrats face: being the party of a passionate resistance but not over the top in a way that would turn off potential voters. Democrats worried Tuesday that methods like public shaming could fuel the flames of an already hostile environment.

"I don't want red gas stations and blue gas stations," said Rep. Gerry Connolly, a Virginia Democrat. "I don't want red restaurants and blue restaurants. We're one country, and we got to try to bring ourselves together."

While empathizing with Waters' frustration, Connolly said her comments resulted in a diversion that took away from the issue of migrant families at the border.

"Here we were getting the country focused on kids being separated from their parents as a matter of border policy, and I think we were making enormous gains in that national debate," he said. "So I don't want to see us distracted by calls to deny retail services to targeted members of the Trump administration. I think that's unhelpful."

House Speaker Paul Ryan offered his own criticism of Waters during a Republican news conference on Tuesday, calling on the congresswoman to apologize. Waters, however, declined to respond to Ryan when asked by reporters about his comments and about criticism by fellow Democrats.

"You know what I've decided? I've decided I'm just talking about the children," she said Tuesday afternoon. "I want the children released. I want a plan. I want a plan for what this administration is going to do to connect these children."

Rep. Cedric Richmond, a Louisiana Democrat who's the chairman of the Congressional Black Caucus, released a statement Tuesday saying Waters was "exercising her constitutional right to freedom of speech" and encouraging others to do the same. Richmond argued the real scrutiny should be on the White House.

"But, for some reason, too many folks find it easier to question the motives of the congresswoman who is protesting the unjust policy, instead of the President and administration officials who are implementing it," he said.

Last week, a group of activists heckled and shouted at Homeland Security Secretary Kirstjen Nielsen at a Mexican restaurant in Washington. A few days later, White House spokeswoman Sarah Sanders was asked to leave a restaurant in Virginia.

"I don't agree with that, frankly," said Democratic Rep. Juan Vargas of California. "I think anyone should be able to go and eat at any restaurant and be able to do it in peace."

Vargas was part of a small group of Congressional Hispanic Caucus members who recently shouted at the President and held up signs as he walked through the Capitol. Vargas argued that his form of protest -- while an unusual scene -- was appropriate, given that Trump was at the Capitol on official business.

"If he were at a restaurant with his family, I wouldn't interrupt him at all," Vargas said. "I don't think that's appropriate. But if he came here to the Capitol specifically to talk about this issue, of course I think it's fair game."

Democrats repeatedly told CNN on Tuesday that going after Trump administration members in private settings was a bridge too far, no matter the reason for the outrage.

"There's too much anger and too much hostility between the parties, and we need to start working with each other more and treat each other more civilly," said Rep. Tom Suozzi, a Democrat from New York.

Many of them faulted Trump for the existing divisiveness and encouraged Democrats to stay above the fray, as they see it.

"I think we have to struggle to remain above that cesspool that we have sunk into," said Rep. Hank Johnson of Georgia, a member of the Congressional Black Caucus.

"I can't tell you just how angry folks are because of the child separation issue," he added. "But cooler heads and calmer heads must prevail."