A) Activate international roaming on your Verizon account. This will allow you to turn off airplane mode and make an emergency call if you need to. However, you'll run the risk of getting a flood of high priced text messages (if you receive a lot of messages) when you turn off airplane mode. If you decide to go this route, make sure to turn mobile data completely off in the settings. You don't want to run the risk of something connecting in the background that you don't know about that might cost a small fortune.
B) Purchase a pre-paid SIM card from a local carrier in the country you'll be visiting in Europe and put it in your phone. This will provide you with a local telephone number and give you an allotment of minutes and texts to use while there, including for emergencies. You can even get data if you want. The advantage with this is that it will be much cheaper than making calls via your Verizon account with international roaming. Prepaid cards abroad can be found in most places for virtually nothing, especially if you don't need a lot of minutes on it. This also gives you the option of calling people or places in the country you're visiting if you need to (for directions, reservations, etc.), or to have people call you. You also don't run any risk whatsoever of incurring any charges from Verizon since the phone will be activated on a European network for the time you're there. And, if for some reason you want to make a call from your Verizon number, if you've enabled international roaming, just turn off the phone, put your Verizon SIM back in, boot it up, and you're good to go.
Personally, I think option B is definitely the way to go. This is what I always do when I'm out of the country. It's really easy, it gives you a working phone in case of an emergency, and it provides a way to contact people locally (as well as a way for them to contact you) while you're in the country you're visiting while avoiding any large charges from Verizon.