"Right now we are focused solely on building a new bridge, not on calls for a new name," Brian Conybeare, Cuomo adviser on the project, told United Press International in a statement.
But the local official who came up with the idea said he was undaunted.
State and federal agencies recently started a $3.9 billion project to replace the decaying 59-year-old Tappan Zee Bridge, the state's longest bridge at 3 miles, carrying an average of 138,000 vehicles a day across the Hudson at its second-widest point.
The eight-lane replacement bridge -- which owner New York State Thruway Authority identifies simply as "New New York Bridge" -- is expected to be completed in 2018.
The supervisor of a town on the Tappan Zee's east side wrote to Cuomo and to local state legislators Tuesday proposing the naming idea.
Seeger, an activist folk singer and seven-decade Hudson Valley resident, died Monday in New York City at 94.
"If the Tappan Zee Bridge is named for Pete Seeger we will honor a man who led the fight for a cleaner river," Greenburgh Supervisor Paul Feiner wrote. "And, we will be inspired whenever we cross the Hudson to think of ways to keep the river clean and beautiful."
"A world-class new bridge should be named for a world-class environmentalist who made our region a better place," the letter said.
Two online petitions on Change.org calling for the same thing garnered some 6,000 names between them in four days.
"Because Pete Seeger's music IS a bridge, from where we are to where we need to be," wrote petition signer Patricia Williams of Alton, N.H.
For nearly 50 years, Seeger sought to highlight the river's pollution and worked to clean it. In 1966, he founded the non-profit Hudson River Sloop Clearwater organization with his wife, Toshi, to protect the river and surrounding wetlands and waterways through advocacy and public education.
The group is known for its sailing vessel, the sloop Clearwater, and for an annual music and environmental festival on the riverfront.
In 1966, Seeger also wrote "My Dirty Stream (The Hudson River Song)," which speaks of his hope the river may someday "run clear."
"The Hudson River is cleaner today because of the incredible work of Pete Seeger and the organization that he helped launch, Hudson River Sloop Clearwater," U.S. Environmental Protection Administration regional Administrator Judith A. Enck told UPI in a phone interview Friday.
"Pete was a regular presence at countless meetings devoted to cleaning up the Hudson River," she said. "His leadership was extraordinary."
Feiner said he was not discouraged by the response from Cuomo's office to his bridge-naming idea.
"The [public] reaction so far is enormous," he said, and the momentary brush-off "gives us time to highlight that there's a lot of support for this."
Naming the new span after Seeger "makes the bridge exciting," he said.
2014: The Year in Music [PHOTOS]