Does the space complexity of a recursive algorithm depend on the total no of recursive calls?
Yes, that's right.

The statement "space complexity is $f(n)$" means that, if we had only $f(n)$ memory and no more, it would still be possible to run the algorithm.

As an example, say we are generating all permutations of length $n$ recursively.
Then we make $c_1 \cdot n!$ recursive calls but, at every moment, use at most $c_2 \cdot n$ memory.
A hypothetical machine with $c_2 \cdot n$ memory, and no more, would still run the algorithm correctly.

The space complexity depends on the structure of recursive calls, not simply on their number.