AUSTIN, Texas—Texas Gov. Greg Abbott on Monday asked hospitals to delay elective medical procedures, as rising Covid-19 cases bring hospitals in many parts of the nation’s second-largest state to capacity.

In Austin, city and county officials used an emergency alert system over the weekend to text and call residents with the warning that the hospital situation is critical, as the number of available intensive care unit beds in a metro area of more than two million residents dwindled to single digits. In Houston, hospitals are into surge capacity and local officials tracking the city’s wastewater system are finding evidence of higher-than-ever levels of Covid-19.

Hospitals have been trying to reduce nonessential treatments and procedures in recent weeks, as Covid-19 cases have rapidly risen, but haven’t eliminated them entirely, as they did during parts of 2020.

The upswing in Texas Covid-19 cases mirrors that of other nearby states as the Delta variant of the virus surges through southern states with relatively low vaccination rates. Texas, which has 29 million residents, confirmed nearly 17,000 new Covid cases Saturday, according to state data, a single-day number not seen since February. The state ranks in the bottom third of the country for vaccinations, with 46% of its population vaccinated, according to Johns Hopkins University.

In Arkansas, Gov. Asa Hutchinson said in a tweet late Monday afternoon the state had eight intensive-care beds available after the largest one-day increase in hospitalizations. “Vaccinations reduce hospitalizations,” the governor said. Roughly 43% of the state’s eligible population is fully immunized, according to Arkansas Department of Health data.

Also on Monday, President Biden discussed with Mr. Hutchinson the increasing case rates in Arkansas and potential additional federal government help, while commending the governor’s efforts to spur more vaccinations, the White House said.

William McKeon, chief executive of Houston’s Texas Medical Center, a group of 60 hospitals that comprise the world’s largest medical district, said the current surge of Covid-19 patients has risen far faster than previous ones and he believes it will get worse. The number of Covid-19 hospitalizations has increased to about 9,500 this week, approaching a previous high of about 14,000 in January. People in their 20s are on ventilators, Mr. McKeon said.

“ ‘We have never seen them sicker when they walk through the door. We have never seen them younger. And they just keep coming.’ ”

— William McKeon, CEO of Houston’s Texas Medical Center

“All of [the hospitals] are saying the same thing: We have never seen a surge like this before,” he said. “We have never seen them sicker when they walk through the door. We have never seen them younger. And they just keep coming.”

Deaths, which tend to lag behind hospitalizations by a few weeks, are a fraction of those seen in January. The state logged 42 deaths July 23, the most recent data available. The highest number of deaths logged in a single day was 387 on Jan. 13.

Texas Medical Center’s hospitals are hundreds of beds into their surge capacity and, while they have additional beds, they don’t have enough nurses to support the hundreds of new Covid-19 patients being admitted each day, Mr. McKeon said. Staffing levels have dropped as some medical workers, affected by the trauma of the last year and a half, have left the field. Texas recently stopped sending state-funded support staff into hospitals in crisis, as the virus appeared to be waning. But on Monday, Mr. Abbott announced the state would begin doing so again, using out-of-state staffing services.

While Covid-19 vaccinations are highly effective at preventing hospitalizations or death from the virus, they’re not foolproof in preventing infection. This poses problems for events like the Olympics and raises broader questions about immunity in the long term. Photo: David Crigger/Bristol Herald Courier/Associated Press The Wall Street Journal Interactive Edition

Houston, the country’s fourth-largest city with 2.3 million people, has developed a sophisticated system of tracking Covid-19 in the city’s sewer system, through weekly testing at 38 city wastewater plants. Tracking began during the first major surge of the virus, in July 2020. Last week, it measured 320% of the Covid-19 levels seen then—the highest, by far, since the pandemic began.

“The wastewater is a bellwether,” said Loren Hopkins, chief environmental science officer for the Houston Health Department, who said Covid-19 usually registers in wastewater a week or more before a person tests positive for the virus. “The numbers really are exponentially increasing.”

The wastewater data can be more accurate than testing data, because it registers Covid-19 being shed by people who may be asymptomatic or never get tested, Dr. Hopkins said. Local health officials are using it to target where to send educational and vaccine outreach, and are doing sampling at specific manholes to track Covid-19 rates in schools.

Mr. Abbott has prohibited local jurisdictions from implementing many Covid-19 related restrictions. In recent weeks, the Republican governor issued a new executive order prohibiting all Covid-19 related limitations on business or masking requirements. Previously, some limited restrictions had been allowed in areas where Covid-19 hospitalizations exceeded manageable levels.

Some school districts and other local governmental entities are preparing for a fight. School districts in Austin and Dallas have announced they will require students to wear masks in the coming school year, in defiance of Mr. Abbott’s order. San Antonio Mayor Ron Nirenberg said Tuesday that he and County Judge Nelson Wolff, the elected executive of Bexar County, had filed a lawsuit against Mr. Abbott seeking to obtain local control of Covid-19 measures in order to protect schoolchildren.

Renae Eze, a spokeswoman for Mr. Abbott, said the governor has the clear authority to overrule local governments on matters of disaster response. “Texans have learned and mastered over the past year the safe practices to protect themselves and their loved ones from Covid, and do not need the government to tell them how to do so,” Ms. Eze said. “Vaccines are the most effective defense against contracting Covid and becoming seriously ill, and we continue to urge all eligible Texans to get the vaccine.”

A district court judge temporarily sided with the San Antonio officials Tuesday afternoon and issued a restraining order against Mr. Abbott’s executive order until a follow-up hearing. The city said earlier in the day that, if granted a temporary restraining order, the county health authority would order masking in public schools there.

Write to Elizabeth Findell at Elizabeth.Findell@wsj.com