It is not the first time that the Donald J. Trump State Park has become a flash point, however. In 2010, the park was on a list of 58 state parks and historic sites that were set for closing because of budget cuts.
At the time, Mr. Trump bristled at the prospect, saying, “If they’re going to close it, I’ll take the land back.” Officials in Albany said they would not give the park back. Ultimately, all the parks and historic sites were kept open anyway.
On Wednesday, the state parks office declined to respond to Mr. Squadron’s request for a name change. Officials in the Cuomo administration who did not want to be identified speaking about what they considered a sensitive topic said removing someone’s name from a state property because of political considerations — whether a park, a bridge or a highway — was a slippery slope.
While declining to give a public statement, the governor’s office referred a reporter to remarks Mr. Cuomo made on Monday about Mr. Trump on CNN.
In the CNN interview, Mr. Cuomo, a Democrat, said that Mr. Trump was “fanning the flames of hate” with his comments about barring Muslims from entering the country. “One billion Muslims were just alienated with one sentence,” he added.
Mitchell L. Moss, a professor of urban policy and planning at New York University, said that the state park should never have been named for Mr. Trump. “Public parks should be named with care,” he said. “We should not go the route of naming public infrastructure or parks after individuals who have yet to contribute to the public sector.”
Seeking a name change for an institution because of the views of the person it is named after is not unusual. Last month at Princeton University, some students demanded that the university change the name of the Woodrow Wilson School of Public and International Affairs because of Wilson’s racist views.
Wilson served as Princeton’s 13th president, from 1902 to 1910, before becoming president of the United States. Members of Princeton’s Black Justice League are now calling attention to his racist legacy, including his observation that “segregation is not humiliating, but a benefit.”
Mr. Trump’s comments have become a lightning rod as well. On the change.org petition, many signers — including people in Westchester County and New York City, but some from as far away as Idaho and Alabama — have cited his remarks during the presidential campaign.
“I grew up in Yorktown and my parents still live there,” Rebecca Capua of Brooklyn, who signed the petition, wrote on change.org. “It was bad enough having this park named after him before he ran for president. Why don’t we name it after Mussolini or someone less terrible?”