Would zirconia crown fill my gap between the teeth? (photo)

you can go with the crown for space closure. but the best option is space closure by orthodontic treatment. Crowns sometimes may look big and bulky.

I would recommend orthodontic treatment to address the deep bite and close the gap without removing any tooth structure. Crowns or bonding are an option. But you have to take into consideration the deep bite. Any restorations you have placed are more likely to fail if the bite is not addressed first. 

Hello.  The first option for you is do orthodontic to close the gap between the teeth and avoid  shave your  teeth, the second option is do dental bonding or veneers instead of do crown in zirconia. 

You may want to consider doing resin bonding or possibly orthodontics to close the gap.  In my opinion, the zirconia crown may be aggressive.  

You can absolutely close the gaps in your teeth without braces. Zirconia crowns would not be my first choice-  as they are not the most esthetic material available.  You also may not need crowns. This correction can be made with veneers. It would be much more conservative to your natural teeth while still solving your problem.