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: $\not\vdash \lnot Pr(\ulcorner 0=1\urcorner)$. Now, the proof of the existence of a provability predicate seems to require much more than
pure logic.