Floyd, a black American man, died after a white policeman kneeled on his neck for several minutes.
His death has sparked eight days of protests in US cities, some of which have turned violent amid accusations of further police brutality used against demonstrators.
It has also seen athletes, sports teams and leagues expressing solidarity with protesters demanding an end to systemic racism and police brutality in the United States.
“Our voices carry, we have a platform and we touch many people,” Boateng, whose father is from Ghana, told Deutsche Welle radio on Thursday.
“All white sportspeople who don't speak out are obviously not racists, but of course it is desirable that they also use their notoriety for this cause. Many do it, but I think we can do much more.”
Boateng added that a deeper message needed to be sent, beyond the world of social media.
“You need to take things in hand, whether in the form of work with children, or in integration projects,” he said.
“That always depends on the parents and what they're teaching their children.
“Also in schools, you need the issue of racism to be integrated in curriculum. It's only by doing that that we will progress.”
People across Germany, including footballers, have been protesting against racism and police brutality in the US over the past days.
More demos will take place around the country this weekend, including one at Berlin's Potsdamer Platz which officially has 1,500 registered participants.
On Tuesday, German Foreign Minister Heiko Maas said that the protests in the US are “more than legitimate.”
Taking a knee
Borussia Dortmund players also showed their support of the Black Lives Matter movement by taking a knee at a training session on Thursday.
Dortmund's Jadon Sancho and Achraf Hakimi were cleared of wrongdoing by the German FA earlier in the week after revealing the message “Justice for George Floyd” on an undershirt during Sunday's win at Paderborn.
Centre-back Mats Hummels posted the photo of 28 players kneeling in the shape of a heart on a playing field on Twitter.
“We the players of Borussia Dortmund fully support the Black Lives Matter Movement,” he said.
“We don't accept racism of any kind. For an open minded and tolerant world, for a better world,” he added.
Earlier on Thursday, Hummels' former Germany and Bayern Munich team-mate Jerome Boateng called on more famous white sportspeople to add their voices to the chorus of outrage against the death of Floyd.
On Monday, the Premier League's Liverpool published a similar photo at their Anfield home.
Made famous by former NFL quarterback Colin Kaepernick's kneeling during pre-game renditions of the US national anthem, bending down on one knee has become a symbol in the fight against racial injustice.