The midfielder collided with Bertrand Traore in the final minutes of the first-half and landed badly, forcing him to be withdrawn at the break.
His teammates went on to lose by a single goal after they failed to find a way back from Gary Cahill's 10th-minute opener, in which the centre-back rose higher than Grujic to head home at the Pasadena Rose Bowl.
Issuing an update on the Serbian's status in his post-match press conference, Klopp explained that the "biggest problem" for him "in this moment" is not the result and rather that "Marko Grujic is in hospital because he can't remember anything."
The Reds boss insisted that's "not too cool for a young boy in this situation" but hoped that the No.16's problem "is not too serious."
Klopp acknowledged that injuries after "games like this in pre-season" are "the biggest problem obviously" as he admitted "it's unlucky" and "not too good for us."
Klopp: Klavan unscathed and Matip nearing return
But the German was at least positive that debutant Ragnar Klavan is unharmed despite being caught late by a poor Cesc Fabregas challenge which saw the midfielder handed a straight red card.
The Estonian defender, the only outfield player to last the full 90 minutes, has not sustained any lasting problems from the tackle, according to Klopp.
There was also good news on Klavan's fellow centre-back Joel Matip, who has now missed the club's last two friendlies after an ankle injury forced him off in the final stages of a win at Wigan Athletic earlier this month.
However, Klopp has confirmed that the former Schalke man is nearing a return to the first-team.
"I expect him to start the season [fully fit]," he told reporters, adding that in order to do so he still needs to "train a little bit more."
Klopp explained that Matip's "first two-and-a-half weeks" with the club were "really good" and he "became better and better" before suffering ankle bruising which left him out.
He noted that it has been "a week or more [that] he has been out" but said Matip was "in a training group" in the hours before the game and that "hopefully he can be back in team training as soon as possible" as he concluded: "The season start at this moment is not in doubt."