"I had to make a selfish decision, but this selfish decision was what was right for my family -- and that is to be here, to be with you guys, and continue to do what it is we've done," he said Monday.
Michigan State is looking for a new coach after longtime coach Mark Dantonio announced his resignation last week.
Fickell, who completed his third season as the Bearcats' coach in 2019, is 26-13 at the school and 2-0 in bowl games.
Colorado coach Mel Tucker also had been considered for the Spartans' opening, but he announced on Saturday that he is staying with the Buffaloes.
Dantonio, 63, said last week that allegations made about him and his football program in a recent lawsuit had "zero" impact on his decision and that he wanted to spend more time with his wife and children after 13 years as head coach of the Spartans.
His announcement came one day after former Michigan State recruiting director Curtis Blackwell filed an update in an ongoing lawsuit claiming that Dantonio and the Spartans committed multiple NCAA recruiting violations. Blackwell said Dantonio helped arrange jobs for multiple high-level recruits and took Blackwell on recruiting visits, which is against NCAA rules because Blackwell was not an on-field coach.