Quitting is not an option, how should one deal with a bully in an office environment when he is close to the owner of the company?

The best thing to do, is ignore!! don't argue with a fool cause you might be mistaken as one! it's bushiness so keep it professional and do your job! and don't sweat the small stuff and never let it get into you, be proud of what you do so no one and nothing can effect what you do best which is your job!

Well.
As you can leave him but he will not leave you, so its your diplomatic skills and political behavior with such a person which will keep you pleased in the environment.
One should NOT compromise on his/her performance, professionalism and integrity for the sack of keeping good but unethical relationships with the bully.
You may consult with your PR/HR Manager if the unfavorable conditions persists.