The band had to pull out of their KROQ Weenie Roast headlining slot at the Irvine Meadows Amphitheatre in Irvine, California, on May 14 and their planned performance for iHeartRadio on May 17 after the singer was rushed to hospital by ambulance with "extreme stomach pains" last weekend.
The 54-year-old drummer revealed the medical emergency was "really scary", but Kiedis (53) was now getting better.
Speaking to UK station Radio X, Smith said: "His stomach was killing him (and) It was really scary. So we went directly to the hospital and we were really upset we couldn't play for the fans, but also more worried about him. I really appreciate all the well-wishes. People sent so much stuff."
Smith also confirmed that the Red Hot Chili Peppers will return to the stage for their scheduled gig in Ohio this weekend and will be good for their headline slots at T in the Park and the Reading and Leeds Festival.
But he admits he and bandmates Flea and Josh Klinghoffer were very worried about Kiedis when he got sick, due to rock star deaths this year with the likes of David Bowie, Prince and Motorhead's Lemmy.
"I know everyone's freaked out when they hear about another musician going into the hospital. It's been such a rough year. He's a very strong man and he'll be back better than ever."
The group's new album, The Getaway, is set for release on 17 June.
"It's super funky and extremely melodic and we just can't wait for people to hear it," Smith said.
