The London-born actor – best known for playing councillor David Horton in the hit BBC sitcom – "peacefully passed away" at 6.45am on Monday 10 January, according to his son Josh.
French, 64, shared a picture of Waldhorn with a broken heart emoji on her social media accounts.
All Creatures Great And Small star Sam West, 55, tweeted: "Sad news about Gary Waldhorn. He played my dad, the eponymous king, in both parts of Henry IV at the Old Vic in 1997 (here with my actual dad as the ultimate non-father, Falstaff). A lovely man and a terrific actor. RIP."
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He shared an image of himself together on stage with his father Timothy West and Waldhorn.
Sad news about Gary Waldhorn. He played my dad, the eponymous king, in both parts of Henry IV at the Old Vic in 1997 (here with my actual dad as the ultimate non-father, Falstaff).
A lovely man and a terrific actor. RIP. pic.twitter.com/7lkzc1GOmF
— Samuel West 💙 (@exitthelemming) January 11, 2022
Waldhorn's son Josh said: "Classically trained, it was the theatre where he truly flourished and he leaves a legacy of entertainment that saw him frequent the boards of Broadway, the West End and our living rooms on the telly!
"He leaves behind his two grandsons, Cooper and Bayley and his son Josh. We will all miss him terribly."
Waldhorn's other work included The Sweeney, Brush Strokes and Hotel Babylon, and most recently appeared in Christmas special The Vicar of Dibley in Lockdown.
His work with the Royal Shakespeare Company also included appearing in Much Ado About Nothing, and he provided voiceovers for adverts for Green Flag, Kwik Fit, Branston and Marmite.
A statement from BBC Comedy read: "Gary was an incredibly talented actor who is fondly remembered by us for his numerous appearances on some of the nations most treasured BBC shows, and of course as Councillor David Horton in The Vicar of Dibley.
"Our thoughts are with his family and friends at this difficult time."
The Vicar of Dibley ran for two seasons between 1994 and 1998, and produced 10 special episodes as well as six Comic Relief offerings.
It was revived in December 2020 with a series of lockdown specials featuring the stars on Zoom calls.
The sitcom has lost a number of other stars in recent years, with Trevor Peacock who played Jim Trott dying in March 2021 and Emma Chambers, who starred as Alice Tinker, passing away in February 2018.
Watch: Vicar of Dibley's Trevor Peacock died last year