What does the US Constitution say about homeschooling?

In the United States it has long been recognized by the Supreme Court that parents have a right to determine the education of their own children.

Supreme Court cases 100 years ago established that all American parents are legally entitled to direct their child’s education, and these cases are still brought up in courts today wherever dispute about homeschooling occurs.

Although the constititution does not directly address education or homeschooling, this right has been decided in the 1920s court cases as a fundamental right. All fundamental rights are considered protected by the constitution under the Due Process clause.

Fundamental rights are a group of rights that have been recognized by the Supreme Court as requiring a high degree of protection from government encroachment… Laws encroaching on a fundamental right generally must pass strict scrutiny to be upheld as constitutional. 
Source: Cornell Law School

So while all Americans have the constitutional right to home educate, regulations and requirements throughout the different states can be quite different from one state to the next. Some states are highly regulated in terms of any qualifications parents might to have to be a home educator, and especailly about reporting requriements, testing and curriculum. Other states are very easy to homeschool in due to the fact that requirements are minimal or even non-existent.

Some statest categorize homeschooling alongside the rules that pertain to private schools and so by considering home education the same as attending a private school, which operates outside the regulations of public schools, exemptions from compulsorily attending public school becomes legal.

Three states that now treat homeschooling the same or similarly to private schooling are Texas, Indiana and California. So regulations that exist for private schools in those states, many or which are not accredited, also apply to home education. In these cases, homeschooling essentially becomes a form of private schools. Some home educators will educate children from outside their home as well, but you should be aware of any local regulations before considering this.

Then there are some states which have statutes that specifically address homeschooling, even if it is not referred as that term. These statutes will describe any requirements that home educators must follow. Some states with statutes relevenat to home education include Iowa, Virginia, New Hampshire, South Dakota and Maine.