YORKTOWN, N.Y. — Donald J. Trump, who is the leading Republican contender for president, is known for splashing his name across everything: casinos, office buildings, golf courses, apartment towers and hotels.

For the most part, even his detractors have grudgingly accepted this as one of the brash real estate developer’s peccadilloes. But after making inflammatory comments about women, Mexican immigrants and, most recently, Muslims, some critics are demanding that his name be removed from one very public place: a state park.

The park is in Westchester and Putnam Counties in New York, straddling the Taconic State Parkway. Called (no surprise) Donald J. Trump State Park, the property was donated to the state in 2006 after Mr. Trump’s plan to develop a golf course there was derailed by environmental and permitting roadblocks.

Since then, the state Office of Parks, Recreation and Historic Preservation has made virtually no investment in the 435-acre park, which has no dedicated maintenance staff and, unlike most Trump properties, exudes a feeling of decay and abandonment. Despite a sign stating that the park is open from dawn to dusk, a kiosk at the entrance was empty on a recent visit — no trail map and none of the usual warnings about ticks and Lyme disease.