It's a little weird but whatever. He's not harming anyone and it's his business. Disagree and no.
It may not be immediately obvious, but this behavior is quite damaging to professional women. Imagine if you could not have a confidential meeting with the Vice President or any other business leader who follows this rule because "you might tempt him," but men in the same position have an open door to access. Pence' behavior may seem like it's all about him, but it perpetuates the glass ceiling for women.