Marshall suffered a wrist injury on a hit by Meriweather, who led with his helmet into the Chicago Bears receiver during Sunday's game.
"I respect the league trying to better our game," Marshall said following Chicago's 45-41 loss. "Guys like that, maybe he needs to get suspended or taken out of the game completely."
Meriweather, who has a history of making helmet-to-helmet hits, also was penalized Sunday for leading with his helmet into the chin of Chicago receiver Alshon Jeffery.
The NFL is expected to suspend Meriweather for either one or two games, a league source told ESPN NFL Insider Adam Schefter.
"I'm not venting or anything," Marshall said. "I understand big hits. That's part of our game, but when you have a guy that does it week in and week out, that's when it becomes a problem."
Meriweather acknowledged that he expects to receive fines for his hits on Marshall and Jeffery.
"Anybody else want to chip in on my dinner?" Meriweather said. "I can't afford it right now."
Meriweather, a seven-year veteran, insisted that he has changed the way he tackles, claiming that he is attempting to go lower to avoid "dirty" hits.
"I feel every hit I took was a legal hit," Meriweather said. "I wasn't trying to be dirty. I wasn't trying to hurt nobody. I didn't lead with [or] lunge with my head. I used my shoulders like they told me to do."
"I know it's not intentional," Redskins coach Mike Shanahan added.
Marshall empathized with Meriweather but said that the defensive back needs to make bigger changes in his approach to tackling.
"I understand, I was one of those guys," Marshall said. "I played defense growing up. I was a headhunter. On the offensive side, I even play that way. But with the rules in place, you have to respect it.
"Guys like that just don't understand. Those are the guys that are in trouble -- as far as [they] don't really have anything to do after football because they think it's all about football. Guys like that ... it's tough."
Meriweather was fined $42,000 for his hit on Lacy, who suffered a concussion on the play. Meriweather also sustained a concussion later in that game when he led with his helmet on another tackle attempt.
Meriweather also was fined $50,000 for a hit in 2010 and accrued $45,000 in fines the following season.
"If you watch my first five years compared to the way I played the last three or four games, everyone in the league will tell you I have changed the way I hit," Meriweather said. "I'm not lunging into people. I'm actually squaring them up. I'm trying to tackle the way I've been coached.
"Am I being targeted? I don't want to say that. I would hope not."
Meriweather said he's not sure what the right way is to hit a ball carrier.
"No matter what I do, I feel I'll be in the wrong," he said. "If I hit you in the shoulder and slipped up, they still say it's head-to-head. If I hit them too low; I think somebody just got flagged for hitting somebody too low. It all depends on who's watching. I don't think we can be right.
"I think they're trying to be safe and I think the only way to be safe is to do what they're doing. But at the same time, this is tackle football. A job of a safety is to instill fear, and you can't do that [by] pulling up."
ESPN.com Bears reporter Michael C. Wright, ESPN.com Redskins reporter John Keim and The Associated Press contributed to this report.