If an employee calls out sick to work & uses a sick day & then the employer suspects or has proof that employee was not actually sick can employer fire the employee??

Depends on company rules, but if they call in and say they are sick and lie, normally yes they can be fired.

If they have "sick" days that are really just personal days, you call in and just tell them you need to take a personal day

What does the time off policy say? Is this a union shop?

But CA is an at will state. An employee can be terminated without any reason at all.