No, Tua hasn't been named the Falcons starter — the QB competition is still very much open. Kevin Stefanski has committed to giving both Tua and Michael Penix Jr. equal reps during training camp, rotating them period-by-period and drill-by-drill to determine a starter (from the Tua Tagovailoa page).
That said, Tua has taken the bulk of first-team snaps during May OTAs and minicamp while Penix works back from his third ACL surgery (still limited to 7-on-7s). Analysts and betting markets lean Tua's way — DraftKings has him at 58% to start Week 1, and ESPN's Louis Riddick has said "Tua should win this job" if he stays healthy (from the Tua Tagovailoa page). But officially? No decision yet. Expect clarity by late July.