"Remy, I hope you're recovering well. It was a physical game and it wasn't my intention to hurt you," he wrote on Twitter.
"I'm also gutted that I didn't get to see u after the game and u weren't well for me to visit u in hospital this morning before we left but I hope to catch up soon brother (sic)."
The high shot resulted in a penalty from fresh-faced referee Luke Pearce - no yellow card.
Earlier in the half, French lock Paul Gabrillagues was yellow carded for a high tackle far less dangerous than the Tu'ungafasi tackle.
The All Blacks went on to score two tries while France tried to defend with 14 men, then their momentum continued for the remainder of the game.
France coach Jacques Brunel felt Tuungafasi deserved to see a card for his illegal shot on Grosso.
"The injury of Remy Grosso is quite serious," Brunel said through an interpreter after the test. "I don't know precisely what the problem is but he is at the hospital.
"I think that the way he was done by the All Blacks pair was illegal."
Brunel said of the performance of his team, who led 11-8 at halftime: "The first half was good… for the second half … the yellow card [for Gabrillagues] was key.
"It was very hard to fight with the All Blacks after that.
"Yellow card or not yellow card, we had to deal with it," he said.
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