I was wondering is a california nursing home contractually obligated to keep a patient for a certain amount of time?

No: However by state regulation under the federal guideline, each nursing home must provide the license for the total number of Medicare and Medicaid beds as well as for the private beds. So for example if the person paid privately but ran out of money and switched to Medicaid. Then depending on the number of Medicaid bed available, this person may have to move another one