Bevin is challenging Senate Minority Leader Mitch McConnell (R-Ky.). So far, his campaign has gained little traction against the Senate's top-ranking Republican.
In the interview, Bevin reiterated his opposition to gay marriage and said the issue should be left up to the states, not the federal government.
"I'm a very strong supporter of marriage as that between a man and a woman; that marriage should be defined as such," said Bevin.
In a statement later, a spokeswoman rejected the idea that Bevin was comparing gay marriage to incest.
"This is a gross misrepresentation of what Matt said," said Bevin spokeswoman Rachel Semmel. "He sees no comparison between gay marriage and incest. He was discussing the implications of the legal rights related to this issue such as hospital visitations and benefits. To imply otherwise is ridiculous."
The Kentucky primary is May 20.