Yes, it will go before the court. I don't think anyone said anything different.
So, Mom wants to relocate. I dunno..let's say Hawaii because it's currently one of a teeny handful of less-than-freezing US states at the moment.
Mom files with the court.
Dad is served.
Now we have a hearing on the matter (for the sake of argument, we'll assume that Mom has included a revised parenting plan suggestion that gives Dad most holidays and agrees to pay the plane fare back and forth).
Dad still doesn't want the move to happen.
Since in CA the presumption is in favor of relocation, Dad has to convince the court why the move should not be allowed.
Example A:
Mom & Dad have "joint custody", with Dad having every other weekend and one night per week with the child.
Dad's reasons for objecting: Child goes to school in CA, and child has friends and relatives living close by. Unfortunately Dad works long hours in the ER, so child-care would be something of an issue.
Mom's reasons: She likes the weather and her Aunt Hedwig lives there.
In this scenario, Dad will likely end up seeing the child next time school is out.
Example 2.
Mom & Dad have a literal joint arrangement with one week on/off.
In this example, Dad's case has virtually been made for him; there are few courts (even in CA) willing to change a true 50/50 timeshare if the reason is to relocate. There are a couple of notable exceptions, but they're unusual enough to make them moot in this discussion.
Example 3
Dad = similar circumstances to #1.
Mom = identical circumstances to #1.
The one hiccup is that Dad is terminally ill. So while ordinarily Mom would be good to go, the court is far more likely to deny permission based purely on Dad's shortened life spam.
So there we have 3 examples, with 3 likely outcomes.
The only reason I wrote this post, AJ, is because you initially stated that if Dad doesn't agree Mom ain't going anywhere (or rather, the child ain't going anywhere). The point being, outside of unusual circumstances Mom is going to "win" every time unless Dad has some wicked good tricks up his sleeve.