The west London side, nicknamed the Cottagers, sit in the relegation zone and are five points from safety in the Premier League.
Supporters reacted with fury when the team lost 2-1 at the Turf Moor stadium in Lancashire after scoring two own goals on Saturday.
Khan, the 36-year-old son of the club’s owner Shahid Khan, thanked fans on Twitter after the match.
He added: “We can close this gap. As with last January we’ll make multiple signings to improve the squad.”
Twitter user Sam Pimm replied: “Leave my club please.”
Khan responded: “Never. I’ll die at this club. Go to hell.”
The vice-chairman was criticised for the tweet but later claimed he had been subjected to disparaging messages from the Twitter user in the past.
He said this included after the side won promotion to the Premier League.
Fulham returned to the top flight this season after beating Aston Villa in the play-off final at Wembley in March 2018.
It brought joy to Cottagers fans as the side returned to the Premier League for the first time since 2014.
Fulham spent around £100m on additions to their squad in the summer, but have won just three league games and their defeat against Burnley was their third in succession.