List of majors offered by each US College/Univ
The National Center for Education Statistics (NCES) runs a database called IPEDS that contains data about virtually all institutions of higher education in the US (any institution that receives federal financial aid funds is required to report to IPEDS.)  Along with information about graduation rates and student demographics, you'll also find a list of all of the degree programs offered by each institution.  

You should also be aware of CIP codes.  These "Classification of Instructional Programs" codes provide a standardized system of numbers to identify college majors. 

It's possible to download data from IPEDS.  The system is somewhat cumbersome to use and it can be slow, but there is an immense amount of available data and you should be able to get the CIP codes of the programs offered at all of the thousands (over 7,000!) of institutions in the US. If you haven't worked with IPEDS before, I'd encourage you to find someone who is an expert to help you understand the system.   

IPEDS also has a user-friendly "College Navigator" search that can be used to search for a particular college and lookup its data.