The world champions' build-up to the new season was blown off course by an accusation of inappropriate and controlling behaviour levelled at Horner by a female member of staff.
Red Bull's parent company, the Austrian drinks giant, appointed an external lawyer to take charge of the inquiry.
Horner was subjected to an eight-hour quizzing by the lawyer at a secret London location last week, days before one of the most successful team bosses of all time appeared at the 2024 car's launch on Thursday.
Formula One made its first public comments on the saga that has rocked the sport three days before pre-season testing begins in Bahrain on Wednesday.
"We have noted Red Bull has instigated an independent investigation into internal allegations at Red Bull Racing," F1's statement read.
"We hope that the matter will be clarified at the earliest opportunity, after a fair and thorough process.
It added: "We will not comment further at this time."
Horner, who has overseen seven drivers world championships and six constructors titles since joining the team on their arrival in F1 in 2005, categorically denies any accusation of improper behaviour.
At this week's car launch, Horner insisted it was "business as usual" whilst admitting the investigation was "a distraction".
Red Bull were dominant on the track in 2023, winning 21 of 22 Grands Prix as Max Verstappen romped to a third consecutive drivers' championship.