From Augustus's conquest of Egypt to its conquest by the Islamic Caliphate in 641, the Roman province of Egypt would have had sea access to the Indian Ocean. Did they ever try to establish themselves as an Indian Ocean naval power? If not, why? Did they have a maritime trade with Asia?
There was a Roman garrison on the Farasan Islands off of Saudi Arabia during the time of Trajan, but that is the farthest known military presence in the sea. As for why not, I suppose there just wasn't any point. Except for one will considered military expedition under Augustus the Romans never even tried to conquer the southern part of the Arabian peninsula, and they never had designs on expansion in that direction. Commerce in the region seems to have operated perfectly well without military intervention, and the Roman state was on good terms with the various Arabian states of the region.