The battle between Texas lawmakers over a bill that would impose some of the strictest voting access restrictions in the country erupted Monday, with Democrats and Republicans vowing that they would not back down on a highly charged issue that has galvanized both parties.

After Democrats killed one of the G.O.P.’s top legislative priorities with a dramatic walkout Sunday night, Gov. Greg Abbott suggested he would withhold pay from lawmakers as a result of their failure to pass the bill.  Mr. Abbott, a Republican who strongly supported the bill, wrote on Twitter, “No pay for those who abandon their responsibilities,” pledging to veto the budget section that funds the legislative branch.

Leaders of the G.O.P. said they would resurrect their efforts in a special session of the Legislature. Briscoe Cain, the bill’s chief architect in the State House of Representatives, said the walkout could allow Republicans to craft a bill even more to their liking.

Despite the Democrats’ victory on Sunday night, Republicans control both chambers of the Legislature and are likely to pass a voting bill in a special session. Mr. Abbott has not stated when he plans to reconvene the Legislature; he could do so as soon as Tuesday, but he may wait until late summer, when he was planning to recall lawmakers anyway to manage redistricting.

When they take up the bill again, they will have to start from scratch and restart a process that could take weeks — though they could start with the provisions in the bill that died Sunday night or even propose one with stricter restrictions.

He and other Republicans were irritated that the walkout had killed not only the voting bill, but also several other bills important to the caucus, such as bail reform.

The bill’s failure to pass was a significant setback for Republicans, and one of the few setbacks they have suffered nationally in a months-long push to restrict voting in states they control. G.O.P.-controlled legislatures in Georgia, Florida, and Iowa have passed new laws with broad restrictions in response to former President Donald J. Trump’s baseless fraud claims.

Many Democrats and voting rights groups considered the Texas bill to be the harshest of all; among other provisions, it would have prohibited both drive-through voting and 24-hour voting; imposed new restrictions on absentee voting; and granted partisan poll watchers broad new autonomy and authority; and increased punishments for mistakes or offenses by election officials.

President Biden condemned the bill over the weekend, calling it a “attack on democracy,” and urged lawmakers to pass two Democratic voting bills that have been stalled in Congress — a theme that Texas Democrats picked up on Monday.

Republicans in Texas and other states that have passed new voting laws have defended them by claiming that they will improve “election security,” despite the fact that the results of the previous election have been confirmed by multiple audits, lawsuits, and court decisions.

Democrats stymied the bill late Sunday night by secretly staging a walkout in the House of Representatives, which prevented the chamber from forming a quorum. As the midnight deadline for passing legislation approached, and with more than a half-dozen Democrats absent, Republican leaders in the House admitted they lacked the necessary number of lawmakers to hold a legal vote and adjourned the proceedings.

Despite promises to bring the bill back in a special session, Republicans were clearly surprised by their failure. They accused Democrats of abdicating their responsibility to govern by walking out — “it shuts down the business of the House,” Mr. Krause said — but also engaged in some finger-pointing within their own caucus. As the deadline approached, Lt. Gov. Dan Patrick chastised House Republicans for mismanaging the calendar. The church was chosen as a deliberate nod to provisions in the bill that Democrats considered to be among the most egregious — those that targeted voters of color.

Democrats said discussions about a possible walkout began as early as April and gained traction as the May 30 deadline for passing bills approached.

When Republicans moved to limit debate in the House, Democrats saw walking out as their only option. When Republicans attempted to move the bill toward passage, all but five of the 67 members of the House chamber left, according to a text message from Mr. Turner.