Several bills have been introduced on state and federal level against transgender students, but none of them says teachers would examine students.
Biden's bill was a departure from the Trump administration's position, which said that that schools may exclude students from athletic teams based on their identities. After Biden's bill, many state legislatures introduced bills that would prohibit transgender students from being part of teams that don’t match their sex assigned at birth. Legislation in Iowa and Georgia contains specific provisions about doctors’ exams. Utah state Rep. Kera Birkeland, a Republican, said "the physical required for sports participation will not change at all" under her bill.
In January 2021, in Congress, Rep. Greg Steube, R-Fla., reintroduced a bill (he had introduced this bill in January 2020 too) that would require schools to interpret Title IX to recognize student-athletes based on their sex assigned at birth. However, the bill does not describe how schools should determine what someone’s sex assigned at birth is.
However, none of the bills stated that a "teacher can do a physical examination for a student". Some bills would open up the possibility for doctors to examine students before they can participate in sports.