Does owning a gross easement give you the right to determine the permanent opening or closure of gates crossing the easement?

An easement in gross as you refer in your question is the right to use another person's land for the benefit of the land that does not have the easement recorded against.

The key to answering your question is the language of the specific easement, its location, scope, length of existence and purpose.

The dispute you mention centers around leaving the gates to the property closed at most times at the request of the property subject to the easement. It sounds as though the properties dealing with the easement are in the country with livestock (cattle) running on the land.

Another factor is custom and practice in the area where the land is located as having gates left open, the use of cattleguards (which essentially prevent livestock from crossing) which allow access via vehicle without the need to open and close gates or perhaps the use of an electric solar powered gate where when you approach the entry point on the neighbor's property, you can trigger a device inside the car to open the gate without the need to get out, open and close the gate.

Hopefully your question was answered and some practical tips to eliminate one getting out of a vehicle to open and close the gates helps in resolving your concerns with your neighbor. Potentially you and your neighbor could agree to pay half of the electric gate or cattleguard as a way to compromise as suggested.