Stuart has made 20 appearances in all competitions for Richmond this season, scoring three tries.
The 25-year-old left Harlequins in 2014, having played three Premiership games for the club.
Meanwhile, scrum-half Sam Egerton, 25, has left the Falcons to focus on a career in law.
Former England Sevens player Egerton made four Premiership appearances for Newcastle this season, but did not start a league game.
Newcastle Falcons have signed former Harlequins scrum-half Sam Stuart from Championship side Richmond in a deal starting from next season.