Can I sue my company for not promoting me?

If the course of action regarding a promotion in your company is to have a vice president sign off on it, then implicit in your immediate superior's "promise" was that your promotion would need VP approval. Therefore you have no legally actionable claim. Things may be different if the VP approved it and in reliance on the promise of that promotion you did something to your detriment. So for example, if the VP approved your promotion but you never in fact received it through no fault of your own; and in the interim you turned down a better position with another company based on the promise, you may then you may have a legitimate claim.

The fact is the without a union agreement, employment contract or company policy to the contrary, you as an at will employee and your employer can increase/decrease your salary/hours, promote/demote and generally mandate work terms/conditions as it sees fit. For your part you can choose to work for your employer or not. Additionally, if no form of workplace discrimination must be present if you are turned down for the promotion. Absent any of the foregoing, your employer has not violated any law.