Background

The issue of discriminating who you sell to in the United States seems to be premised on the fact that businesses are public accomodations according to this article:

Whether you post a sign or not, businesses never have the right to refuse or turn away customers because of their race, gender, age, nationality or religion. In addition to the Civil Rights Act of 1964, several states have their own civil rights legislation designed to prevent discrimination. The Americans with Disabilities Act also prohibits discrimination in public accommodations, making it illegal to refuse service to individuals who are disabled or handicapped.

Given that a Mixed Economy with a preference toward a free market is the economic system of the United States, from a free market point of view, this seems unnecessary to do. There will always be businesses and choice in the market. Businesses that discriminate will put themselves at a competitive disadvantage due to alienating a certain employer base at an extra cost to the business:

Businesses that discriminate based on a host of job-irrelevant characteristics,
including race, ethnicity, gender, age, disability, and sexual orientation and gender
identity put themselves at a competitive disadvantage compared to businesses that
evaluate individuals based solely on their qualifications and capacity to contribute.

According to this Free Market point of view, using government enforcement seems to create a control where there does not need to be one.

Question

Why does the government need to institute a legal policy to prevent selling discrimination?

TL;DR: You question rests in the supposition that, next to the business that refuses to sell to me, there will be another similar business that will be willing to sell to me. That supposition is, to put it mildly, optimistic, and you ignore historical examples of institutionalized/collective racism that disprove it.

You take the POV of an individual discriminatory business owner in an overall non-discriminatory environment without taking into account other competence limiting factors.

But maybe I live at a small town that is served by two pharmacies, and -since racism is often a social convention- both pharmacy owners forbid me from buying medicines. Maybe I have no car or other transport means¹ to use it to access another town with pharmacies easily enough (in the case that those pharmacies are not owned by racists, too). And of course, being it an small town and my race being a minority², the market would not be big enough to support a third pharmacy³ to compete with the already established ones.

¹Or, we go back in time enough, simply there were no cars or other ways of transportation enough that would make travelling to the big city an easy activity.

²And probably not a rich minority.

³And that ignores possible barriers put forward by a racist major/council/public... again, do not think only of individual racism but of a racist community. If my race is 5% of the population and none of the remaining 95% will shop at a pharmacy that sells to me, it would be very difficult for such pharmacy to be profitable.

Other people already pointed out how restrictions in supply of goods will damage the targeted groups no matter what, but in addition to that, the supposition that discrimination is harmful to the supplier is also not necessarily correct.

In a society that is highly discriminatory to certain minorities, a business that bans those minorities might gain much more patronage by the bigoted majority than they would gain from accepting the patronage of that minority.

For example, I'm sure that during Jim Crow businesses that served blacks didn't ban whites, that certainly didn't result in them out competing white only businesses to dominate the market though, bigoted whites self segregated and discriminatory businesses managed to thrive.

According to this Free Market point of view, using government enforcement seems to create a control where there does not need to be one.

The federal government action was to counter-act the state government action of having police enforce southern apartheid, and courts refuse to punish the extra-judicial violence that supported it.

If let's say your local supermarket had a sign "***** not welcome", you are just looking at the "free market" problem.

You are ignoring the real problem: That ***** cannot buy their food and what else they need from that supermarket, forcing them to walk or drive further to do their shopping, possibly paying higher prices, possibly splitting up families (where for example dad is **** but mom is not).

The government should and likely will have laws to prevent people from being hurt by discrimination.

Simply, because discrimination does not necessarily cause companies to go out of business.

Other answers have dealt with the problem of preventing concrete harm to customers who are being broadly discriminated against, which would remain a plausible justification for such laws even if any such business would inevitably fail, since new discriminatory businesses might continue popping up, and have also pointed out how in certain circumstances discrimination may be economically beneficial. However, another point is that even economically harmful discrimination will not necessarily force a business out of the market!

Discrimination is indeed often a poor business decision. It represents forgoing potential income, and most economic analyses suggest that a discriminatory business will be less successful than a non-discriminatory one. However, being less profitable only will lead to failure in a situation of perfect competition, where no differentiation between products is possible. In this case, the non-discriminatory business will spend less money on the same quality on labor, while also having more positive publicity and being able to take advantage of economies of scale. In an ideal situation, this would drive the discriminatory company out of business. And it is probably true that, in the modern United States, less-discriminatory businesses will be more successful.

The thing is, effectively, discriminatory businesses are not only taking on additional costs to meet the prejudice of their owners or managers, but also offering a differentiated product. Let's consider a much less harmful form of discrimination: exclusive clubs. Groups like Mensa charge money to belong to their organizations based on characteristics like scores on IQ tests. They offer some small benefits, yet they aren't outcompeted by a group that offers those same benefits to the general public. Why? Because the exclusivity is part of what they're selling. A group that offered a T-shirt with their name on it or access to their clubhouse to the entire population wouldn't be Mensa. No group offering the same services regardless of IQ scores can price Mensa out of the market, because they're not selling the same thing.

As a more direct example, a golf club that charges a significant membership fee and won't accept Jews or black people, or a neighborhood that limits homeowners to a certain religion (and yes, both these still exist) are selling an environment in which people can be surrounded by others who look and think like them. In a similar fashion, discriminatory businesses are actually effectively selling their exclusivity along with their physical products. A bakery that won't sell cakes to gay customers may well be responding to the owner's beliefs, but from the perspective of the market they're also selling a place where customers don't need to worry about seeing two people of the same sex holding hands. This is compounded by there being correlations between different
things that a company can sell: for instance, Chik-Fil-A isn't just selling an environment free of gay people, but rather a conservative Christian environment generally.