The Briton, Formula One world champion with Brawn GP in 2009, is replacing Alonso at McLaren.
But Button, who returns to the sport after taking a break for 2017, says he is not interested emulating his former teammate.
The 37-year-old says: “Indy’s not really been something that I’ve thought about. Personally, I was surprised that Fernando was interested in doing it but we all like different things.”
Button said he would like one day to compete in Le Mans, another leg of the famed Triple Crown of Motorsport along with Indy and the Formula One world championship, and also NASCAR in the United States.
He adds: “And then there’s the other motorsports that I love like Rallycross as well. There are many things, but Indy hasn’t been up there for me for many different reasons.”
Meanwhile, Lewis Hamilton lapped faster than ever at Monaco as the Mercedes driver and Ferrari’s championship leader Sebastian Vettel set the pace in first practice for Formula One’s showcase race yesterday.
The triple world champion, who can equal the 65 career pole positions of boyhood idol Ayrton Senna in tomorrow’s qualifying, topped the timesheets with a best lap of one minute 13.425 seconds on ultrasoft tyres.