There are two sources in federal law that provide "Due Process": the 5th and 14th Amendments.
5th Amendment:
"nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."
This has been interpreted as saying that no person shall be deprived of life, liberty, or property, by the federal government without due process of law. The NFL is not the federal government.
The 14th Amendment:
"nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws"
This has been interpreted as preventing state governments from depriving a person of life, liberty, or property without due process of law. The NFL is not a state government.