“This is the first big step in restoring fiscal responsibility in Ohio,” said Kevin Bacon, a Republican senator.

The battle in Ohio has unfolded over the past month along with similar confrontations in Wisconsin and Indiana. Unlike in the other states, where Democrats are needed for a quorum, in Ohio, Republicans make a quorum on their own.

In Wisconsin, a standoff over legislation that would cut collective bargaining rights for public employees intensified. Democrats in the State Senate, who left Wisconsin last month to prevent a vote from taking place, learned that the Republicans left behind were taking steps to start fining missing lawmakers $100 for each day they stay away.

Though some of the Democrats and Republicans met early in the week, it was clear by Wednesday that the sides were farther apart than ever. And each side suggested that the other caucus appeared to be fracturing under the tension. From an undisclosed location in Illinois, Mark Miller, a leader of the Senate Democrats, issued a statement accusing the Republicans of “schoolyard bully tactics.” Scott Fitzgerald, leader of the Senate Republicans, then issued his own and reminded the Democrats of Gov. Scott Walker’s warnings that some 1,500 state workers might be laid off soon if the lawmakers did not act on the bill.

In Ohio, Nina Turner, a Senate Democrat, said, “This bill seeks to vilify our public employees and turn what used to be the virtue of public service into a crime.” At its heart, the bill redraws rules governing how several hundred thousand public-sector workers bargain with the governments that employ them. Among the most objectionable parts, workers said, are the rules giving local officials the final say in breaking labor impasses. Currently, non-elected, third parties decide.