My attempt at a proof by induction:
Let two successive primes be $a$ and $b$. $a - b = c$ gives the number of numbers between them.
Base case: The first five values of $c$ are $0,1,1,3$ and $1$. There is no pattern that indicates the next value of $c$ hence they are arbitrary.
Inductive step: Assume the values of $c$ are arbitrary for all $2 \le K \le P$ where $K$ and $P$ are primes. Since, all the values of $C$ till $P$ are arbitrary, any value of $P - K$ will be arbitrary relative to the previous $C$s
Hence all $C$s will be arbitrary.
Is this proof correct?
No, this is not correct. The term "arbitrarily" means that for any $n$, there is a prime gap of length $\geq n$. It doesn't mean that there is no pattern.
There are also some other problems. You didn't state your induction hypothesis. You also didn't prove anything in your base case. In the induction case, you use "arbitrary" as if it's a property of numbers when it isn't.