Fletcher told fans in a video on his Instagram account: "Hi all, I know that a lot of you have seen by now that I tested positive for COVID. Thank you so much for all your messages, I'm fine.
"But it means that I'm obviously isolating, and I can't finish the McFly tour, which I am sorry, so gutted about, it hurts so much letting you guys down, you have no idea."
The 36-year-old – who performs lead vocals and rhythm guitar in the chart-topping boyband – was replaced by Tom Abisgold for McFly's show in Glasgow on Sunday night.
Fletcher went on: "But the shows must go on. You guys have waited long enough for a night of fun McFly entertainment that I know Danny, Dougie and Harry are more than capable of giving you... without me there.
"Obviously it won't be quite as good without me, but it will be a McFly show. If you have tickets for the last two nights, go and have an amazing time, sing your hearts out.
"We have a good friend of ours who is also called Tom, he's standing in for me, he's been a McFly fan since he was like five years old so it's amazing he's getting to stand in and be part of McFly for a couple of nights, so thank you, Tom."
Abisgold will perform with McFly at the final two concerts of the tour in Liverpool on Tuesday 28 September and Nottingham on Wednesday 29 September.
The band are also due to play a one-off gig at London's O2 Arena on 21 November, before taking a break from touring until June next year.
Fletcher and his dance partner Dowden will miss the first dance-off of Strictly 2021 and be allowed to return to compete in Week 3.
He said: "This also means I have to have a week off from Strictly so I won’t be there for week two. I’m sorry to all my Strictly team, fellow contestants and to my brilliant partner Amy, sending you loads of love Amy.
"I will be watching on Saturday cheering you all on guys, I can’t wait to see your week two dances. I’ll be so jealous that I’m not there. I’ll be back for week three – Amy, I’m gonna dance my little Cuban heels off to hopefully get more than a four from Craig [Revel Horwood]
"Until then, I’m going to rest and recover. Once again I’m so sorry to everyone and I’ll see you soon."
The pair performed the very first dance of the series on Saturday night, doing the Cha Cha to September by Earth, Wind & Fire.
New judge Anton Du Beke tipped them to make it through to the final, telling Fletcher: "A fabulous start – well done you, you've set the bar. Straighten up your leg actions and I think that will see you all the way through, my old sausage."
Revel Horwood said: "I thought it was a little bit wooden, but what you do have is amazing rhythm.
And Motsi Mabuse said: "I think we are going to see you go far. With that start, what can go wrong?"
The All About You singer has previously admitted feeling under pressure given he's following in the footsteps of bandmate Harry Judd, who won the show in 2010.
Watch: Who is Strictly's Tom Fletcher?