Why does a circuit need to be closed for a capacitor to develop voltage?
Current (the flow of electric charge) flows in a loop or circuit.


  Shouldn't the positive terminal of the battery exert an attractive force on the electrons in the wire, 


No. There are no positive charges sitting on the top of the battery. The charge in the battery is balanced. There is an electro-motive force, however, and if a circuit is created charge will flow.


  ... causing a net positive charge to develop on the top plate of the capacitor? 


Nope. Since there is no charge on the top of the battery there won't be on the capacitor.


  And then, because like charges repel, shouldn't the positive charges on the bottom plate get pushed away, thereby creating a voltage in the capacitor? Or will the positive charges not get pushed away because they have nowhere to go unless the circuit is closed?


The last line explains it.





Figure 1. Another water analogy. Source: UWaterloo.

See the linked article for another water analogy.


The battery is represented by a water pump.
Your open circuit would be represented by pinching the pipe below the pump to shut of current flow.
The capacitor is represented by the membrane blocking the flow.


If you pinch the pipe and start the pump the membrane won't budge (capacitor won't charge) because no current is flowing.