Bove secured a release from his contract at Roma this month ahead of making the move to Watford as a free agent.
The 23-year-old suffered a cardiac arrest two years ago while with Fiorentina.
Now fitted with a cardiac defibrillator, Bove has been prevented from resuming his career in Italy due to athletes being barred from competition with such devices.
Bove has now signed a deal with Watford to 2031 and stated: "I have been eager to get back on the pitch, and doing so with a historic club like Watford fills me with immense pride.
"I am excited to train with my new team-mates and to play, and I cannot wait to show my gratitude for the trust that has been placed in me."
Watford's assistant sporting director Valon Behrami also said: "We are delighted Edoardo has chosen us over many options he had.
"Edoardo is a top, proven talent and he is here to help us fulfil the objective of the owner which is to challenge for a return to the Premier League."