Nainggolan limped off after half an hour of Sunday's Serie A clash at the San Siro, where Inter won 1-0 thanks to a late Mauro Icardi header.
The former Roma star appeared to sustain the injury in a heavy tackle from Lucas Biglia that saw the Milan midfielder booked, although Nainggolan also caught the Argentina international in the incident.
Nainggolan initially played on but after he was on the end of another firm challenge he was replaced by Borja Valero.
"Nainggolan has been chopped down," Spalletti said to Sky Sport Italia.
"We have to take him to the hospital and see. He won't be with us for a while."
Both teams had a goal ruled out for offside while Stefan de Vrij hit the post in the first half, but Spalletti was in no doubt the result was fair.
"It's obvious Inter deserved it, we played from start to finish in their half, pressed high and went for it," the coach added.
"Therefore, we deserve compliments. Not because we wanted it more, but because we played better. If you say to me, 'The team that wanted it more got the win', that is not accurate. The team that played better football won the game.
"I came here to win, not to just sit on a bench for a season and bring home a salary. I came here to organise the future of Inter and the players must do the same.
"I liked that after the final whistle, Icardi said we must find consistency and keep pushing. Inter chose us to bring this side back to the level that the fans deserve. It is our job, we must do it with a sense of belonging and prove they were right to choose us.
"It's dangerous when you press high, as there is always the risk of conceding a counter, but we were perfect and forced Milan to continually lose the ball. We ensured Milan were incapable of playing their quality passing game that Gennaro Gattuso had spoken about."