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The victory was the 82nd of Hamilton's career and ninth of the season.
A spokesman for the governing body FIA said the International Court of Appeal in Paris had found the Swiss-based team's protest was not admissible.
The German, whose last victory was in Belgium in August 2018, took the chequered flag 2.641 seconds ahead of unhappy Monegasque team mate Charles Leclerc.
Valtteri Bottas was second for Mercedes with five-times world champion Lewis Hamilton a distant third.
It was also Ferraris first win this year.
Hamilton now has 250 points, with team mate Valtteri Bottas on 188 and Verstappen closing in fast on 181.
Russian Daniil Kvyat was an astonishing third to hand Red Bull's sister team Toro Rosso their second ever podium.
It was Hamilton's seventh win in 10 races this season and 80th of his career.
The Dutchman's victory for the second year in a row at Spielberg also dealt champions Mercedes their first defeat of the season.
The five-times world champion now has 79 career wins, 12 short of Michael Schumacher's all-time record with 13 races remaining this season.