Tell your dad -- loud and clear -- that this device isn't a punishment, it's a gift that will allow him to live independently, in his own home, for longer.
The alternatives are a full-time caregiver, an assisted living facility, or maybe moving in with a relative.
Talk realistically about how, at this stage of his life, he's more medically fragile and at greater risk for medical emergencies. This isn't his fault; it's just nature doing its thing. Point out his strengths -- that he's still able to live alone and take care of himself in many ways.
You may also want to bring up real scenarios based on his condition. What would your father do if he had another stroke while getting out of the shower, causing him to fall, and he couldn't get back up? Or if he had acute shortness of breath and couldn't find the phone? Or if he felt dizzy and weak and was scared to stand?
This may sound like a scare tactic, but these are real possibilities for your father and he needs to see this for himself. Explain that an emergency response system is like a home or car alarm, a practical and reassuring tool for protecting things that are valuable and dear.
If your father is still hesitant, get in touch with your PERS supplier for suggestions. The company may have other clients he could talk to or give you ways to modify the transmitter button to make it more comfortable for your dad.