Ben Crump urged Americans to step back and take a "broad view" of the underlying issues.
"I think we have to figure out how to honor people who have done things that are beneficial to society, and if they did things that were not beneficial to society, that we can examine in the lens of having a broad view of what we believe as Americans represents the best attributes of our national identity, then we should look at that," Crump told Fox News host Neil Cavuto on Saturday.
"Whether it should be a situation where, if we keep statues up like that, we tell the history of that individual so people will know the whole story," he added.
"I'm not sure pulling the statues down is the right thing if we now don't get the lessons to understand how we can learn from those things, so we don't repeat those mistakes of the past. You know, they say history -- if not studied -- we will often repeat it."
Crump, whose office is in Houston, represents several black families who have lost loved ones at the hands of white police officers. Floyd died on Memorial Day while in police custody after an officer pressed his knee on the back of Floyd's neck despite his pleas that he couldn't breathe. The officer, Derek Chauvin, has been charged with second-degree murder.
Across the U.S., protesters have vandalized and destroyed Confederate monuments, as well as figures of others with no ties to the Confederacy, such as George Washington.
The night before Crump's comments, protesters toppled a statue of senior Confederate officer Albert Pike in Washington, D.C..
His remarks touched on widespread concerns that removing objectionable material from the public square could effectively leave Americans unprepared to fight future injustices. In an apparent effort to prevent that, HBO has said it will resume streaming "Gone with the Wind," but only after including relevant context.
Former Defense Secretary Robert Gates has suggested that Confederate monuments should be placed in museums rather than being displayed "in places where it appears that we're celebrating them."
During the Fox interview, Crump also commented on the Rayshard Brooks case, which involved another black man's deadly encounter with police, this time in Atlanta.
"These are very contentious times when it comes to the whole issue of police use of force," the Houston lawyer said.. I think ... we should all try to make sure that we examine all the attendant circumstances.
"I hope that's what the district attorney did, because I've always advocated that we have to have due process of the law for everybody and that we want to have the fair administration of justice for all involved, and what I have found is that normally, the police are going to be given every benefit of the doubt, every benefit of consideration."
Crump added: "The thing that I have found is that citizens of color are not always given that same due process, and so I think it should be fair for everybody. It should be fair for police officers just as much as I believe it should be fair for citizens whether they are black people, white people, brown people or red people.
"We have to make sure that when we quote Thomas Jefferson's words that all are created equally that we honor those words in America by saying we want equal consideration and equal justice under the law for all Americans."