I know that Islam looks at men and women in an equal light. However, I have heard that in Islam, it is not permissible for a woman to become the ruler of a nation? Is this the case, and if so, why?
This is not the case. In Islam, the election of the head of state is based on the vote of the majority. Whoever enjoys the confidence of the majority whether he is a man or a woman is legally eligible for this post.
However, the only debate which may remain is that whether women in general are suitable for this job regarding their temperament and nature. Nevertheless, if the majority does elect a woman for this post, no one has the authority to veto the opinion of the majority.