The LCR Honda rider crashed braking aggressively into the first corner having started 12th, wiping out both the Ducati and the Suzuki riders.
Rins, who slammed MotoGP's stewards for taking no action on the incident, was left with a left wrist fracture, while Nakagami was kept in hospital overnight having hit Bagnaia's rear wheel with his face.
The Japanese rider says he has "no fracture and I am OK" after spending a night in the Hospital General de Catalunya in Barcelona.
He also apologised to both riders for the accident, adding that it was "all my fault."
In a statement from his team, Nakagami said: "First of all, I want to apologise to Alex Rins and Pecco Bagnaia for yesterday's accident. It was all my fault. I wish Rins a speedy recovery and I hope he is in good shape for the German Grand Prix.
"After yesterday's crash, I feel lucky because I have no serious injury, no fracture and I am OK. I spent the night at the Intensive Care Unit, under observation but all is good. So, I'll try to be ready for the next race in Sachsenring."
The crash came a week after Nakagami and Rins made contact at Mugello, with the Suzuki rider branding the Honda rider "one of the most dangerous".
Sunday's incident prompted much criticism from other riders, much of which was directed at the Stewards' decision not to penalise Nakagami.
Rins called for the Stewards' panel to be changed, saying they were "not at MotoGP level," while Franco Morbidelli said the decision was only good for the show and not the riders.
Bagnaia, whose title hopes were dented by his retirement, said he was "disappointed" with what happened, and branded it "not acceptable", while race winner Fabio Quartararo also felt the crash should not have been deemed a racing incident and urged riders to be "conscious" of the dangers in first-lap encounters.
Pramac's Johann Zarco said Nakagami "lost all his credit" with the riders who backed him after his Mugello clash, which was brought up in Friday's safety commission at Barcelona.
MotoGP returns at the Sachsenring from 17-19 June.
