Yes, it is ok to do this, but you are asking for collaboration, I think. The other person may not have the time or energy for it, of course.
It is safe with almost everyone, but there is a certain amount of poaching of ideas in academia, so be sure the professor has a reputation for ethical practice.
I would send an email stating that you have a draft of a paper that touches on some of the ideas of the professor and ask if he would be willing to collaborate on its completion and publication. Don't send the paper initially, but say something about it. The working title might be enough if it is descriptive, but you might want to include something about the direction of the research.
By the second email, make sure that you have worked out the authorship question if that is important to you.
Note, that by raising the issue you also raise an expectation that you will share your current work. It might be embarrassing to back out, though it might be necessary depending on how the communication goes.