Short Circuit vs. Complete Circuit
My question is, is there not current flowing into the battery in a complete circuit? 


Yes, but it is limited by the load resistor. The resistor is so-called because it resists the flow of electricity.


  How come the battery is unable to handle the current under its own voltage? 


The battery will have some internal resistance - in fact, we usually model them as an ideal voltage source with an internal series resistance - but the series resistance value in a good / new / healthy battery is usually very low. That means that in the short-circuit condition the current can be much higher than is safe for the battery. Think what would happen if you short out your car battery positive to the chassis with a spanner!


  And if some form of resistance on the circuit is required in order to prevent a short circuit scenario, how much of a resistance do I need, in order to prevent the battery from overheating? 



Read the battery's specification from the datasheet and work out what the safe current is.
From Ohm's Law we can calculate the minimum resistance as $$ R = \frac {V_{batt}}{I_{max}} $$


e.g., A 12 V car battery can supply 40 A for several minutes.

$$ R = \frac {V_{batt}}{I_{max}} = \frac {12}{40} = 0.3 \ \Omega$$



Incidentally, the power being delivered in the above example is given by \$ P = VI = 12 \cdot 40 = 480 \ \mathrm W \$.





Figure 1. (a) The apparent LED key-fob lamp. (b) The more correct model showing the cell's internal resistance. Source: LEDnique.

On a related note, the internal resistance of a button cell is used in key-fob LED lamps to limit the current to a safe value for the LED.