No, you can't omit 'van' is this sentence. However, when asking for the time, you can get away with not using a verb.
- Milyen nap van? = "What day is it?"
- Ma kedd van. = "Today is Tuesday."
- Hány óra [van]? = "What time is it?"
- Tizenkét óra [van]. = "It's twelve o'clock."
The other question regarding the time, "Mennyi az idő?" doesn't have a verb either.