There's no such thing as a director's role in the sense you're thinking of. The closest role is a creative director but that's not just a role you waltz into as a beginner and is a lot more than just directing. It differs from studio to studio but taking this job posting as a starting point, it requires that you:
Again, all positions will vary but this one requires previous experience, leadership experience, financial experience, managerial experience, experience in leading art/design teams specifically and leadership positions in 3-5 previously shipped AAA titles (6-15 years). A producer role is a more realistic aim but even then, you'll almost certainly need previous experience as a producer on a title or previous experience in leadership positions on art/design teams.