"If I had a son, he'd look like Trayvon," Obama added.
That comment drew criticism from Republican presidential hopefuls, who accused the president of politicizing the shooting.
"Is the president suggesting that if it had been a white who had been shot, that would be OK because it wouldn't look like him?" Newt Gingrich said on Sean Hannity's radio show. "That's just nonsense. Dividing this country up, it is a tragedy this young man was shot."
Rick Santorum leveled similar criticism during an appearance on the Hugh Hewitt radio show.
"Politicizing it, this is, again, not what presidents of the United States do. What the president of the United States should do is try and bring people together, not use these types of horrible and tragic individual cases to try and drive a wedge in America," Santorum said.
Cutter dismissed the criticism, saying Gingrich and Santorum were trying to manufacture controversy in the "last, desperate" moments of the GOP primary campaign.
"They should listen to their own words," Cutter said. "Nobody doubts the president's commitment to bringing the country together."
She went on to tell the GOP candidates "they should watch their words."
Cutter is not the only Obama aide to push back against the GOP criticism.
White House senior adviser David Plouffe called Gingrich's comments "reprehensible" and said the Republican presidential hopeful was "clearly in the last throes of his political career."