The U.S. government wants a federal appeals court to let it enforce new rules restricting abortion-related counseling services by recipients of federal family planning grant money while the rules’ legitimacy are determined.

The new regulations govern Title X, the only federal program devoted to ensuring people receive family planning care and counseling. They prohibit a recipient from performing, promoting, referring for, or supporting abortion as a method of family planning. The rules also require physical separation of abortion and all other services offered by grantees.

The Title X program offered $286.5 million in grants in 2018. It serves about four...