A group of workers at Google parent Alphabet Inc. has asked the company to stop collecting data on users seeking information about abortions, according to a petition sent to Chief Executive Officer Sundar Pichai this week.

The petition, signed by more than 650 workers, also called on the company to remove search results for crisis pregnancy centers, which it said were misleading to those seeking abortions. Crisis pregnancy centers are typically nonprofits that counsel women against having abortions.

Representatives from the Alphabet Workers Union sent the petition to Mr. Pichai on Monday after circulating it among employees last week, a spokesman for the union said. Mr. Pichai hadn’t responded by the end of the workday on Wednesday, the spokesman said.

A Google spokeswoman declined to comment on the petition. Alphabet has reported that it had 174,014 full-time employees at the end of June, in addition to a large number of contract workers.

Tech companies and location-data brokers are under heightened scrutiny after the Supreme Court ruling in June that overturned Roe v. Wade. Privacy advocates fear that prosecutors will use warrants or subpoenas to demand data revealing users who have visited abortion clinics or sought out related information. That data could be used to build legal cases against people who are charged with having an abortion in states that have outlawed the procedure.

Google, like most tech companies, has said it responds to lawful requests from government agencies for user data and that it pushes back against requests that it deems overly broad or otherwise objectionable.

Google’s handling of abortion-related policies has become a politically charged issue. Before the Supreme Court ruling, more than 20 congressional Democrats urged Google in a letter to Mr. Pichai to take action limiting the appearance of crisis pregnancy centers in abortion-related searches.

The following month 17 Republican attorneys general responded with their own letter warning that they would take action against the company if it suppressed results related to crisis pregnancy centers, which the group said provide important medical services.

The Alphabet union in late June released a public statement asking Google to stop storing “any data that could be used to prosecute users in the U.S. exercising their bodily autonomy.”

Google said in July it would begin automatically deleting data on physical visits to abortion clinics logged by the company’s products.

Mr. Pichai said in an email to employees following the announcement last month that Google would “work on new ways to strengthen and improve these protections over time.”

The petition this week pressed the company to go farther. It asked that Google introduce “immediate user data privacy controls for all health-related activity,” such as searches related to reproductive issues, and to stop saving any user information related to abortion services. It also called on Google to fix “misleading search results related to abortion services by removing results for fake abortion providers.”

Among its other demands, the workers asked Alphabet to extend certain reproductive-healthcare benefits to contractors, end lobbying efforts through its in-house political-action committee and take steps to limit advertisements on publishers of “disinformation related to abortion services.”

Workers who signed the petition asked Alphabet to create a task force with 50% employee representation for handling abortion-related issues across the company.

Write to Miles Kruppa at miles.kruppa@wsj.com