United has daily non-stop flights from Houston (IAH)

American Airlines has daily non-stop flights from Miami (MIA)

Other airlines may have season flights from other US airports.

Some non-direct flights can go through San Pedro Sula (SAP) or Tegucigalpa and then take a local carrier to Roatan (RTB)

Reputable Local Carriers are:

CM Airlines

Aerolinas Sosa