Mr. trident is arguing that the hospital has violated his 14th amendment right to due process. what part of the 14th amendment is he referring to in his argument?

Here's the part of the 14th Amendment Mr. Trident is referring to:

No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law.

History/context:

The 14th Amendment guaranteed rights to life, liberty and  property to all citizens born in the United States -- including those who formerly were slaves in states in the South.  The amendment was ratified in 1868, following the Civil War, and was one of three amendments (XIII, XIV, XV) which were aimed at making black Americans full citizens of the USA.  The full text of the 14th Amendment stated:  

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; 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 would be a true statement