A public good is a good that has two characteristics. First, a provider cannot prevent a given person from using it. Second, when one person consumes the good, it does not detract from others' ability to consume that good.
Looking at this definition, it is possible to argue that higher education is not a public good while elementary education is such a good. Access to higher education is much more limited. Higher education costs much more to provide than elementary education does. Therefore, when one person consumes the good, it is much more likely to detract from others' ability to do so as well. America has the resources needed to provide elementary education to all who want it. But higher education is too expensive for the country to be able to provide it to everyone. Therefore, there is some amount of competition for spots in higher education. This means that it is not a public good.
Public good is free for everyone to consume. Elementary education is considered as public good in many countries. It is the right of every citizens to received elementary education from the government. At this young age, people are highly dependent on others just like they are depending on their parents to feed them. On education, parents are responsible for their children's education but they have to depend on the government to provide it.
However, it is not the same for higher education. One thing, people have a choice for higher education. They can choose to majoring in many different field. The choice might be for economic-related reason or for any other reasons towards their advantage. Logically, it is not right for the public to support a very high cost of education for somebody to make a good future income. Therefore, it makes sense for higher education to be a private good.