A prominent white nationalist who supports Donald J. Trump has started a smear campaign against Evan McMullin, the independent candidate who is running close with Mr. Trump in Utah, with a robocall accusing him of secretly being a homosexual who supports open borders and amnesty.

The robocall, which the white nationalist, William Johnson, narrates and is being made across Utah, demonstrates how concerned Trump supporters are that a Republican could lose the state for the first time since 1964. It is also the latest example of Mr. Trump attracting support from fringe political activists who are shunned by most mainstream Republicans.

In a deeply personal attack, Mr. Johnson says in the call: “Evan has two mommies. His mother is a lesbian, married to another woman.”