My project manager is pushing me to get it done before I go on
vacation, but it's unlikely.
How do I deal with the project manager?
As always, when there are differing expectations between you and your project manager - you talk.
Find a time to talk this through with the project manager.
Explain you understanding regarding the remaining work, and the time you have to get it done before leaving for your permitted vacation time.
Discuss what you think needs to be done, and how long you estimate it will take
Discuss other alternatives if the work exceeds the time available to complete it before vacation
Discuss how you will pick up the remaining work after vacation, if that's the chosen path
Discuss how you can hand off some of the work to others for completion during your vacation, if that's the chosen path
Discuss altering your vacation plans, if that's what you choose
The key here is - discussion. That's how you should deal with most problems.
Did I make a mistake by booking my vacation off?
There's no way for us to know if this was a mistake or not, since we don't know the norms, rules, and expectations of your company.
Again - discuss this with your project manager to determine if you made a mistake or not. Knowing this could prevent similar mistakes in the future.