No, Market Grill is in the main arcade.
From pikeplacemarket.org :
Fish Sandwiches, Platters, and Clam Chowder
The Market Grill serves the finest fish sandwiches available. Choose from salmon, halibut, cod or prawns. Market fresh fish, vegetables, and bread are purchased daily in the Pike Place Market. Our clam chowder and seafood stew are Seattle favorites.
Location: In the Main Arcade across from Pure Food Fish
Hours: Daily from 9 am to 6 pm
Address: 1509 Pike Place #3
Seattle, WA 98101
Clarification: East, across the "street," (Pike Place), fronting on First Avenue, is the Pike Place Bar and Grill. Back West across Pike Place is the Market Grill. They could not be more different. The "counter with 3 sides" is a great description for the Market Grill, and the little quotes around 'restaurant' make me think Christy hit it. The sandwiches are fairly plain, but grilled well, tasty, and served barely out of the flow of pedestrian traffic. Very small menu, on a chalkboard or smallish sign, I believe.
Yes, Lowell's is very near, but probably has a small telephone book for a menu. Close by Lowell's (which I, a 56 year native who grew up in the market have never visited), is The Athenian. Culture on the half-shell. Have a hang-town fry with the old-timers.