Great question! Gravity is indeed a major force that influences tides, but not the only one.

Newton postulated that the tides are related to the gravitational attraction between the earth and the moon. His law of gravitation (summarized and without some values) states that the force of attraction between two objects is proportional to the product of their masses and inversely proportional to the square of their distance (that is:
d*d or d², where d is the distance).

The result of this is that if the moon and/or earth were larger and closer together, the gravitational attraction would be larger.

However, while tides are based on this principle,tidal generating forces are actually inversely proportional to the cube of the distance between the objects (that is d*d*d or d³, where d is the distance). Tidal generating forces are calculated by taking into account the gravitational attraction of the earth, moon, and sun and also the rotation of these objects. Therefore, tidal generating forces are more dependent on the distance between the objects than the size of the objects (like in the gravitational force). This is why the moon dictates the tides of the earth, not our sun. The National Ocean and Atmospheric Administration has crunched the numbers:

"Although the gravitational attraction between the Earth and the Sun is more than 177 times that between the Earth and the Moon, the moon dominates the tides. Our sun is 27 million times more massive than our moon. If tidal forces were based solely on comparative masses alone, the sun should have a tide-generating force that is 27 million times greater than that of the moon. However, the sun is 390 times further from the Earth than is the moon. Thus, its tide-generating force is reduced by 390³, or about 59 million times compared to that of the moon. Because of these conditions, the sun’s tide-generating force is 27/59, or about half that of the moon (Thurman, H.V., 1994)."

sources: here, and here.