Well, in the State that I am licensed the answer would be NO (in your present situation). You would need to take course work in a Civil Engineering discipline probably leading to a Masters Degree in Civil Engineering to qualify for the PE exam. Now let me say that all state Boards are NOT alike, so it may be easier in some States versus others. Again in my State you only receive ONE PE license and license number. One could become a licensed SE with a PE, but that is it.
In most States, a Professional Registration Board reviews your credentials related to your engineering discipline prior to approving you to take the exam. You would also need to demonstrate specific work experience in CE, as well.
As a side note, I have seen engineers obtain multiple licenses – I would guess for different States. Say for example you obtained a PE license to practice ME. Now you decided to go to graduate school and obtain a masters in Structural Engineering. After x years of experience you apply to the State Board to sit for the SE exam's to become a licensed SE.