Heart of Atlanta is the first in what became a significant number of landmark cases that dismantled "Jim Crow" laws in the United States.
The motel in this case argued that the Civil Rights Act was an unconstitutional interference with its Fifth Amendment right to run a private business as it saw fit.
The Court rejects this argument, saying Congress has the authority to regulate interstate commerce. The Court doesn't base its decision on the "civil right" you have to stay at the motel.

So why did the Court base its decision on the Article I Commerce Clause power rather than simply saying that a person has a right to public accommodations regardless of race under the 14th Amendment to the Constitution? Wouldn't that have "balanced" the economic right and the social right at issue here? Cite your sources