HOUSTON — The Republican-dominated Texas Legislature inched closer on Monday to passing some of the toughest abortion restrictions in the country as the State House of Representatives approved a bill that would ban abortions after 20 weeks of pregnancy and hold abortion clinics to the same standards as hospital-style surgical centers.

Advocates for abortion rights said the measures would force dozens of abortion providers around the state to close and leave only a handful in a few major cities that are able to meet what they described as excessive, costly and medically unnecessary restrictions. Opponents of abortion, including Gov. Rick Perry, Lt. Gov. David Dewhurst and other Republican leaders, said the legislation was aimed at protecting women’s health and unborn children.

After the State Senate passed a version of the bill last week, Mr. Dewhurst, who has called for overturning Roe v. Wade, the landmark Supreme Court decision that legalized abortion, posted a message on Twitter celebrating passage of the bill, known as Senate Bill 5. He wrote: “We fought to pass SB5 thru the Senate last night, & this is why!” Attached was a graphic produced by opponents of the bill that said the legislation would force many clinics to shut their doors.

A few hours later, Mr. Dewhurst attempted to play down his original post, saying that both he and the bill were “unapologetically pro-life” and for women’s health. But opponents said his first message demonstrated that the legislation was an unconstitutional attempt by Republican leaders at a backdoor statewide ban on abortion. In January, Mr. Perry told those gathered at an annual anti-abortion rally in Austin that his goal was to “make abortion, at any stage, a thing of the past” and that the ideal world was a world without abortion.