ALBANY—Thousands of gun-rights advocates rallied Thursday on a muddied lawn outside the Capitol here, protesting firearms restrictions passed last month by the Legislature and signed by Gov. Andrew Cuomo.

The protest drew about 5,000 people in a light rain, including the president of the National Rifle Association, David Keene, who warned that Mr. Cuomo's policies on guns, same-sex marriage and abortion would cripple any potential bid for the White House.

"What he's doing is the mirror image of what Republicans seeking the presidency were doing in the last cycle. What he's doing is seeking measures that would appeal to a very small segment of the left wing of the Democratic base," Mr. Keene said of Mr. Cuomo, a Democrat. "And if he gets that far, he's going to find himself the most unelectable person nationally."

Mr. Cuomo's assumed ambitions were a motif on Thursday. One man hoisted a sign on the Capitol steps. "Cuomo's White House Check List," it read. "Kill 2nd Amendment. Kill babies in 9th month. Kill up-state jobs. Kill NYS."

There was little chance the demonstrators will get what they want: repeal of the gun laws. Mr. Cuomo and legislative leaders say they won't make changes beyond "technical corrections" to a bill that expanded the state's ban on some semi-automatic rifles, created new registration requirements and stiffened penalties for some gun crimes, among other things.

For his part, Mr. Cuomo, who hasn't declared a presidential run, stayed miles away from Albany on Thursday, traveling instead to Brooklyn and Long Island. But he addressed simmering anger about the gun measures.

"This is not about taking anyone's gun," he told reporters, calling the legislation "a reasonable approach that basically keeps guns out of the hands of criminals and people who are mentally ill."

National and state polls show that majorities support new gun laws after the Newtown, Conn., school shooting. Reaction to New York's measure has been most vociferous upstate, where some were already uneasy with the governor's policies.

Mr. Cuomo has come under criticism from some officials and residents for delaying a decision on hydrofracking, a form of natural-gas drilling.

"Look at upstate New York, there are no jobs here," said protester Marc Podolski, 42 years old, from Highland, N.Y. "If you opened up fracking, you open up thousands and thousands of jobs."

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