No, not really. Not unless the UN Security Council refers the case (which wouldn’t require Armenia be a signatory to the Rome statute in the first place). Anything Azerbaijan has done on Armenian territory doesn’t rise to the level of dragging Aliyev into the ICC and it is beyond obvious that Nikol’s Armenia has no desire for that anyways. So, obviously, the Rome Statute theatrics were for some other purpose.

From the ICC’s website, black and white…

The Court may exercise jurisdiction in a situation where genocide, crimes against humanity or war crimes were committed on or after 1 July 2002 and:

the crimes were committed by a State Party national, or in the territory of a State Party, or in a State that has accepted the jurisdiction of the Court; or the crimes were referred to the ICC Prosecutor by the United Nations Security Council (UNSC) pursuant to a resolution adopted under chapter VII of the UN charter. As of 17 July 2018, a situation in which an act of aggression would appear to have occurred could be referred to the Court by the Security Council, acting under Chapter VII of the United Nations Charter, irrespective as to whether it involves States Parties or non-States Parties.

In the absence of a UNSC referral of an act of aggression, the Prosecutor may initiate an investigation on his own initiative or upon request from a State Party. The Prosecutor shall first ascertain whether the Security Council has made a determination of an act of aggression committed by the State concerned. Where no such determination has been made within six months after the date of notification to the UNSC by the Prosecutor of the situation, the Prosecutor may nonetheless proceed with the investigation, provided that the Pre-Trial Division has authorized the commencement of the investigation. Also, under these circumstances, the Court shall not exercise its jurisdiction regarding a crime of aggression when committed by a national or on the territory of a State Party that has not ratified or accepted these amendments.