The 38-year-old moved up from assistant to take over on an interim basis following the departure of Jack Ross and has won, drawn and lost in three games in charge.
Former Dundee United striker Stevie Crawford will return to Tannadice as assistant head coach.
Fox told the club's official website: "I'm absolutely delighted to be given the opportunity to lead this outstanding football club.
"It's a massive honour and privilege and I want to thank Tony (sporting director Tony Asghar) and the board for their faith in me.
"I'm looking forward to putting a team on the park that the supporters can be proud of.
"We have made some small steps in the last few weeks and it's now about getting that winning feeling back.
"I know how much the supporters at this club crave success. They, quite rightly, want a team that is fully committed to the cause every week and it's over to us, as a coaching staff, and the players to achieve that.
"This group of players have real potential and quality. It's down to us to find consistency in our performances to deliver the success that we all want.
"I'm also delighted to have Stevie on board as my assistant. He's a brilliant human being, a real football man and he has real experience in the game.
"I've known Stevie a long time and I have enormous respect for him as a coach. He cares deeply about developing players and teams and will be a real asset to Dundee United."
Dundee United sporting director Tony Asghar said: "We are delighted to announce Liam Fox as head coach at Dundee United and the appointment of Stevie Crawford as assistant head coach.
"Liam was clear in his interview with the board on how he plans to improve short-term performance and results as he outlined a coaching and tactical plan to allow Dundee United to return to our long-term strategy.
"He was brought to the club as one of Scotland's best young football coaches and now it is his time to lead the team to success.
"It is also exciting to appoint Stevie Crawford as assistant head coach. Stevie has vast coaching experience and a strong track record of developing forward players in his previous roles.
"I have no doubt this coaching team will bring much-needed fresh and innovative coaching and game models to Dundee United."