No. Accidentally declining a call or contacting the prison from your end does not put your inmate's phone privileges at risk. Phone privilege loss is tied entirely to the inmate's own conduct, not to actions taken by people on the outside.
In Arkansas and everywhere else, inmates lose phone access for specific infractions they commit themselves. Saying something prohibited on a recorded call, causing a disturbance while using the phone or waiting in line, misusing or damaging the phone equipment, or receiving a disciplinary sanction where phone restriction is part of the punishment are the standard reasons privileges get suspended. None of those involve anything you did or did not do on your end.
Accidentally hitting the wrong button when a call comes through happens regularly and is not flagged as an issue on the inmate's record. The call simply does not connect and the inmate can try again when the phone is available. That is the entirety of the consequence.
If you are concerned about missing calls going forward, making sure your phone is set up to receive calls from the facility's carrier without any barriers is the most practical fix. If collect calls are not working on your cell phone, setting up a prepaid account with the carrier the Arkansas facility uses ensures calls come through reliably. InmateAid's discount phone service is also worth considering if the per-minute rates are running high, as it can reduce costs significantly depending on the facility and your location.