It is normal and natural react you get eyes watery when you are scared. This is because out tear glands can react to the nerves. That is why we tearing when we got some emotion such as sad, painful and moved. So, you needn't take it seriously and it won't cause any hurt of you. As long as you calm from such satiation, you don't tearing again.
Well, I am sorry to hear that since watering eyes would be troublesome at any rate. It is true that some people will have biological reaction to something frightening, perhaps you are one of them, and you will automatically produce water in your eyes. Perhaps you can go and visit a doctor to see if there are any problems with your eyes or something. Personally speaking, it is not very serious, but you should be careful, just in case.
When you get scared or sad, the emotional nerves will be excited. Your eye nerves will be influenced and the tear gland system will work under the extreme scare situation. It is not a big problem. But if you feel discomfort after the watering, you should take notice of it.
That is instinctive action of human beings. Like kids, they would cry when they are afraid.
When we are scared, cranial nerves take immediately action to stimulate lacrimal gland which would produce tear.
It is common. If we were cold blood, we would have nothing to be afraid. But we are human with emotions.