to anyone. You have probably seen these signs in restaurants and other businesses. But in today's world, can a business owner really refuse to serve someone? The answer is it depends.
The Federal Civil Rights Act guarantees: "full and equal enjoyment of the goods, services, facilities, privileges, advantages, and accommodations of any place of public accommodation, without discrimination or segregation on the ground of race, color, religion, or national origin.". The Americans with Disabilities Act extends this to include people with disabilities. So currently, federal law does not extend these rights to sexual preference.
Some states, such as California, have extended the Civil rights act to include sexual preference. It passed something called the Unruh Civil Rights Act which makes it illegal to discriminate based on sexual preference and what it calls "unconventional dress". Other states have done similar things.
The point here is that currently it is a State's rights issue, not a Federal issue. So states are free to adopt the Federal Civil Rights Act as is or they can extend it to include other things, like California has done.
What Tim Cook and others are trying to do is have it become a Federal issue so that the Federal Civil Rights Act is amended to include sexual preference. Much like gay marriage proponents are attempting to do. Then states will be powerless to invoke their own individual standards.
We can debate whether or not it is the right thing to do but at the heart of it is a Federal vs. States rights struggle.