Melk is just an old benedictine monastery and a small village and nothing worth mentioning regarding food.
OTOH Melk is located at the end of the Wachau valley, which is full of great restaurants.
From Emmersdorf, across the river from Melk, til Krems there are dozens of smaller and larger inns, wine bars, restaurants etc., etc...
The most famous are the Prandtauerhof in Weissenkirchen, the Loibnerhof in Loiben or the Jamek in Joching, the best is probably the Landhaus Bacher in Mautern.
For something more down-to-earth we like the Kirchenwirt in Weissenkirchen or the Zum schwarzen Bären in Emmersdorf. But there is so much more to find, just make a small trip along the north side of the Danube river...