First, a remark. You formulate both Gödel's theorem and your question
in a subjective way "we cannot prove", etc. However, this theorem is a
mathematical one, therefore it is not about our ability to do
something, but about the nonexistence of a mathematical object, namely
a formal proof within the system concerned. You can draw some e.g. 
philosophical conclusions from this theorem, but this is a completely
another matter. 

Now, as far as your question concerned, it is almost certain that
nothing analogous to Gödel's theorem can even be stated for the pure
first order logic itself. The reason is simple. The analogous theorem
would claim the unprovability of the formula expressing the the
fact that a contradiction is unprovable _within the pure first
  order logic_.  But, in the absence of the formal provability
predicate, this theorem cannot even be stated. Actually, what we would
like to show is that there is a formula $Pr(x)$ such that, on the one
hand, it can be considered a provability predicate (that is, for any
formula $\varphi$, $Pr(\ulcorner \varphi \urcorner)$ is true just in
case $\vdash \varphi$ (here, of course, $\ulcorner \varphi \urcorner$
is the Gödel number of $\varphi$), on the other hand, the formula
expressing the fact that a contradiction is unprovable is itself unprovable: $\not\vdash Pr(\ulcorner 0=1\urcorner)$. Now, the proof of the existence of a provability predicate seems to require much more than
pure logic.