The singer announced on Tuesday that he was leaving the BBC One dancing competition after three live shows for "personal reasons".
His dance partner Karen Clifton has said she is "so sad" at his departure.
Strictly will proceed as normal this weekend, with a third celebrity eliminated on Sunday.
Young, 37, sent a text to BBC Radio 2 Breakfast Show host Chris Evans after quitting the contest, which was read out on air.
It said: "Here's the thing. I'm absolutely dandy, thanks. It's all very undramatic, really.
"I've done my statement and others can say what they want to say. I never read what they say anyway."
He is only the fifth celebrity contestant to pull out of the show in its 13-year history.
In his statement, the former Pop Idol winner said he was leaving Strictly "with joy in my heart that I have been able to take part in one of the most loved shows on British television".
He said: "To be a part of Strictly has been a long time ambition of mine. As a performer, a viewer, and a fan of the show, to dance as a contestant was an experience I always hoped for.
"I have made some great friends, and am in awe of their performances week in, week out. I have found a creative partnership with Karen that has been the most wonderful thing to experience."
He promised to keep watching as a viewer and wished his fellow Strictly contestants "so much luck".
Young and Clifton came joint fourth on the leaderboard after last weekend's movie week, and the pair had been considered early favourites to lift the glitterball trophy.
They were due to dance the Viennese Waltz to Say Something by A Great Big World and Christina Aguilera on Saturday's show.
Clifton said: "I'm so sad that Will has decided to quit the show and I know it wasn't an easy decision for him.
"It was great dancing with him, he was such fun to work with and I wish him all the very best for the future."
Young had been criticised by head judge Len Goodman for his dance not containing enough salsa elements. When Young disagreed, he was advised to "turn up, keep up and shut up" - but Goodman stressed this was "only a joke".
A Strictly source said there had been no falling out between the pair.
"The pair have a lot of mutual respect for each other and had a laugh about what had happened backstage together when the show came off air on Saturday evening," the source said.
Other celebrities taking part in the show have reacted with sadness to the news.
Daisy Lowe wrote on Twitter: "@willyoung you are such a talented superstar... @bbcstrictly won't be the same without - I love you dearly and hope you're alright."
Ore Oduba said: "So sad to see our @bbcstrictly friend @willyoung leave us. Been amazing sharing the floor with you, even better getting to know you."
Laura Whitmore tweeted a quote, seemingly in relation to Young's departure, reading: "Be kind, for everyone you meet is fighting a battle you know nothing about."
Former Hollyoaks actor Danny Mac said he was "gutted" to lose Young from the show but added that everyone "fully supported" his decision.
And former contestant Jeremy Vine, who had been paired with Clifton - also known by her previous surname Hauer - said: "Can't believe we are going to miss eight weeks of @karen_hauer dancing, one of the greatest dancers @BBCStrictly has ever had, and my friend."
Former contestants John Sergeant, Kelly Brook, Jimmy Tarbuck and Jade Johnson have all previously withdrawn from the show.
Comedian Tarbuck left in 2006 for health reasons, followed by actress and model Brook in 2007 after the death of her father.
In 2008, journalist Sergeant said he felt there was a "real danger" he might win which would have been a "joke too far", and Olympic long-jumper Johnson quit the following year after she tore a ligament in her knee.
This year's series is to continue as planned with an elimination every week until the final.
The contestants who have previously been voted off - DJ Melvin Odoom and EastEnders actress Tameka Empson - will not be returning to the Strictly dancefloor.
A BBC spokesman said of Young: "The show fully respects his decision and wishes him all the best for the future."
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Will Young says he is "absolutely dandy" after leaving Strictly Come Dancing, stressing that his early exit is "undramatic".