No, you don't need a stone floor, wood is enough if you've built correctly.

First of all, you should straighten and consolidate your construction site (with the hoe).
Then you put round wooden beams in the ground, that's a little tricky but it works. Alternatively, you can also use the hoe to adjust the building floor again afterwards.
It is important that every second floor plate is a beam, since the fireplace is then exactly on the round wooden beam, you can then place it on wood.

Only if the wooden floor is not supported is the stability insufficient for the hearth.