I built a simple desk out of 72" butcher block and H frame metal legs. You can move the table side to side slightly. If I add a 1x4 or 2x4 piece of wood to the back of the legs along the bottom, will that stabilize the table? I guess I know it will stabilize it, but will that be enough to stabilize it?
In answer to your question: highly unlikely, your wood on steel connection will not be as fixed as a welded steel on steel one would be, and the resulting structure is not forming triangles either. That combination will continue to make it wobbly.
As mentioned by khrrck in the comments, a couple of brackets and screws into the back wall would solve your problem. As it's just a couple of screws, I wouldn't even consider it a permanently fixated construction. It would take all of 2 minutes to detach if needed.
However, if you're unable or unwilling to fixate it to the wall, a couple of diagonal braces that fixate those steel legs to the wooden surface will go a long way. Preferably four (one in each corner), but two at the back will already help a lot.
At the steel end you can probably get away with drilling a hole through both the wood and the existing leg, and attaching these two components with a long, thick bolt (for example M10). On the other side a bracket with a number of smaller screws (so they do not puncture the butcher block) will probably work best (where the wooden beam is diagonally sawed off so it sits flush on the bottom of the butcher block, for convenience you can use a 45 degree angle, same as the beam as a whole). Assuming you will want the table's surface to align flush with the wall, mount these beams on the 'inside' rather than at the back.
Example diagonal braces in green (smaller in front so they are less in the way) and and example of large X-bracing on the back in yellow.