The Ivy League school said the resignation will take effect on Dec. 31.
Williams took over at Yale in 2009 and listed himself as a Rhodes candidate on his resume. He was referred to as a finalist when Bulldogs quarterback Patrick Witt had to decide if he would go for his Rhodes interview as a finalist or play against Harvard this fall.
The university began an investigation in November after the New York Times reported officials of the Rhodes Trust said they had no records of Williams applying for the fellowship when he graduated from Stanford in 1992.
Williams said in a statement Wednesday he was encouraged to apply for the scholarship by Stanford's fellowship office but acknowledged he never did.
"I considered the opportunity, sought advice and was encouraged to apply by faculty members and my coach Bill Walsh, but I did not apply," he said.
Williams' biography on the Yale football website also mentions that he played for the San Francisco 49ers practice squad in 1993. In his statement, Williams also said he never signed a contract with that team.
"During the spring of my senior year, I was given an opportunity to attend a tryout camp with the San Francisco 49ers, and I participated in that three-day event. I did not sign a free-agent contract with the 49ers or participate in their summer training camp for signed players," he said.
Williams was 16-14 at Yale and 5-5 in 2011. But he also was 0-3 against archrival Harvard.
"I am extremely proud of my academic, athletic and coaching career," he said. "If there was confusion created, I take full responsibility. The timing of this inquiry has been difficult for everyone. At this point I believe it is in the best interest of my student-athletes and Yale University that I step down."
Athletic director Tom Beckett said the search for a new coach would begin immediately.
"Coach Williams has been a dedicated teacher and coach at Yale, and his students have benefited greatly from his leadership and passion for the game of football," he said.Suggest a correction