I believe the manager of a private service can arbitrarily decide not to serve someone. But what about a public service, like a public health clinic?
In normal commercial situations there is the principle of freedom of contract where parties are free to contract with whom they choose. As a result of this principle, they are also free to not choose to contract with whom they choose (i.e. refuse to serve someone). There are limits, such as if it could be argued that by doing so contravenes other laws, such as those against discrimination on the basis of race, age or gender for example.
When a public entity is involved, it is a branch of government and is governed by the rules that define the relationship between the individual and the state - the constitution, so principles of constitutional law apply. Decisions made by public bodies can therefore be subject to judicial review, where an affected party or someone with sufficient standing can take the matter to court to be reviewed by a judge.