Please refer to Pourbaix diagram to find the optimum range of pH values where both oxides are stable. I think instead of adding HCl to make an acid catalyzed gel, you can try ammonia to form a base-catalyzed and obtain stable ZnO-SiO2 phase.

ZnO is amphoteric in nature, hence it usually will dissolve at high or low pH values. You could try a base route instead of the acid route you have used  (as suggested by A. S. Babu) and ammonium hydroxide being a weak basic solution might just work fine. You could also consider tinkering with your synthesis route in such a way that you first produce a stable silica sol before incorporating ZnO particles; that way you may be able to avoid direct contact between ZnO and HCl / Alkali while trying to generate your silica sol.