It really depends.
In Northern climates (where I grow) the rows are much further apart, typically in the 10 foot range. This is because the colder it is, the more important sun becomes as a resource. The closer the rows, the more other rows will shade the other grape vines thus resulting in a smaller overall sun exposure. The spacing between the vines itself is dependent on the vigor of the grape vine. More vigorous grape vines need more space (generally 6ft apart), less vigorous vines are closer together (4 ft apart). However, most commercial vineyards are pushing their vines in the 4ft range now regardless of vigor as it gets the vineyard producing much quicker. If you have vines 4 ft apart and rows are 10 feet apart you will have 1089 vines per acre.
In warmer climates and major wine regions the rows are much closer together. The grapes will typically ripen every season therefore sun isn’t the main factor in plaining row spacing like it is in the North. Cost is. Land is very expensive in the main wine regions, therefore it is very important that you see your rows closer together. Rows can easily be as close together as 4 feet apart. In this example, if your vines are 4 feet apart and your rows are also 4 feet apart then you will have 2723 vines per acre.
If you want to play around with different spacing’s yourself the best thing to use is a calculator. This calculator is for tree spacing but, it works as well for vine spacing.