The female actor is known for starring in the Kannada film Mugilpete and Malayalam period film Pathonpatham Noottandu.
Meanwhile, the US Consulate in Chennai expressed condolences in Kannada and English.
The hospital where the actor was admitted had declared him brain dead on Monday, following which his family decided to donate his organs.
In the health bulletin released by the hospital at 8.20 pm on Monday, the doctors confirmed that Vijay was brain dead.
The actor has also written to Chief Minister B S Yediyurappa to regularise the services of around 300 cemetery workers in Bengaluru, who have no security to fall back on.
For the past 10 days, the 27-year-old Mandya-born actor has been driving Covid-19 patients to hospital and also taking the bodies of those who didn't survive the disease for their last rites
He wrote that he is aware of issues like low wages, lack of leaves and holidays, long working hours and overtime.
Directions have also been sent to inspector-general Arun Chakravarthy to send a team of divers from Panambur in Dakshina Kannada for the search operations.