Magath's men saw their 13-year stay in the top flight brought to an end on Saturday with a 4-1 defeat at Stoke City.
The German arrived at Craven Cottage in February following the departure of head coach Rene Meulensteen, but has only been able to mastermind three wins during his time in charge.
The 60-year-old signed an 18-month contract with the London club and is determined to honour that deal despite Fulham's fall into the Championship.
"We as a team, and I include myself and management, would like to apologise for our bad performance on Saturday," Magath said in an open letter to fans released on the club's official website.
"It certainly was not what you as fans were expecting and should rightfully demand of us, but we were unable to perform on the day, resulting in our downfall.
"I was appointed by (Fulham owner) Mr Khan in February, and willingly signed a contract for 18 months, realising that there was a distinct possibility that we would not survive in the Premier League.
"Contrary to many opinions, I was not only brought in until the end of the season, with the primary target of keeping Fulham in the Premier League, but to build a team that can compete in the top flight.
"The reality is that we have been relegated, but my commitment and desire to Fulham remains.
"In the short space of time that I have spent here, I have been very impressed with so many aspects of our football club, and with this in mind, my one priority is to give Premier League football back to you.
"During the summer months, there most certainly will be many changes within the club, we obviously need to improve and redefine our playing staff. We have in our academy team, talented, young and hungry players.
"On Sunday we play our final game of this season against Crystal Palace, and we owe it to you to put on a good performance that is worthy of your fantastic support.
"I am looking forward to the task ahead and to a successful future for all of us."
