Cuomo supports a bill that would require "micro-stamping" that laser-etches identification marks onto the shell casings of semiautomatic weapons, the New York Post reported.
New York streets are regularly scenes of gun violence.
Cuomo said the shooting had reached "an undeniable point."
"It's hard for anyone to refute the damage that's being done," he said.
A micro-stamping bill passed the Assembly in June but was blocked in the Senate.
New York Mayor Michael Bloomberg, a former Democrat who is now an independent, says he favors the bill, while Senate Majority Leader Dean Skelos, a Republican, says he is open to passing more stringent gun laws.
Gun-rights advocates don't see the need for new laws.
"New York already has the toughest gun laws in the nation -- and clearly, as evidenced in New York City, they are not working," said Tom King, president of the New York state Rifle & Pistol Association.
Notable deaths of 2014 [PHOTOS]