On Saturday, the Coast Guard reopened the western part of the river to vessels that checked with the patrol commander enforcing the safety zone, but it said the eastern half remained closed because of the destabilized bridge.

Coast Guard officials did not immediately respond to a request for comment.

The problem came a day after Mr. Cuomo and other dignitaries gathered at the new bridge to celebrate the planned opening of the second span. Mr. Cuomo lauded the new bridge as a testament to “New York’s greatness” that “shows the world that we’re capable of dreaming big and delivering results for the people of this state.”

The first span of the new bridge, a $4 billion project, opened last year after decades of wrangling over whether to repair the Tappan Zee or build a new bridge.

The westbound span of the bridge had been carrying both eastbound and westbound traffic. Eastbound traffic was set to be shifted to the new span after it opened this weekend, but concerns about the stability of the old bridge delayed that shift.

The new bridge, which connects Rockland and Westchester Counties, is a two-span cable-stayed design, with the cables holding up its steel decks anchored to the tops of the angled central towers rather than to the shore.

The old Tappan Zee was closed to traffic in October and was being disassembled piece by piece. By December, each end of the bridge had been removed so that it did not connect to either bank of the Hudson River.

As the Democratic primary in New York on Thursday approaches, Mr. Cuomo’s opponents seized on the problem with the old bridge, with some suggesting that Mr. Cuomo prematurely announced the opening of the span of the new bridge. The goal had been to fully open the new bridge in 2018, and Mr. Cuomo said last month that he was looking at a mid-September completion.