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The DC Motor is question is here: DC Motor

I was wondering if you use a diode and a switch in a manner shown in the below picture if the motor would exhibit regenerative braking when SW1 is open? (Assuming that the voltage from the motor when braking exceeds the motor power) Is the use of PWM required?

^{simulate this circuit – Schematic created using CircuitLab}

(Assuming that the voltage from the motor when braking exceeds the motor power)

That's the problem — it doesn't. You need a way to boost the voltage coming from the motor to a level that will actually charge the battery. You can use a separate boost converter, or you can create a more tightly integrated solution that uses the inductance of the motor itself as an element in a boost converter. Either way, it does involve some sort of PWM control in order to regulate the power flow.

The voltage on the terminals of your motor will be a function of its speed and the load.
It will not be higher than your power supply, unless some external force is trying to accellerate the motor.

Regenerative breaking supposes that you get back some energy from the motor while breaking. Therefore the breaking must be performed by applying a load accross the motor. For example, you could "short-circuit" it using a low value/high power resistor. The resistor would force the motor to supply power and transform that power into heat. At the same time the motor slows down.

When we use a resistor to slow the motor down, the energy is lost into heat. We could use that heat and transform it into "electricity", but that is not the most effective way.
It is better to change the resistor with a more complex system that would transform the power. It can for instance be an inductor. In way similar to switched power supplies we can "charge" the inductor and "discharge" it into your power source. It would be applied to your power source in a different path than the one controlling your motor through PWM.

So the diode accross the controlling switch would not do anything while breaking your motor.

Your circuit doesn't provide any braking because the motor is running free when the switch is open, and won't produce higher voltage than the battery unless it is 'over-driven' to higher speed by an external force.

To brake the motor you must put a switch across it, like this:-

^{simulate this circuit – Schematic created using CircuitLab}

When SW2 is closed it 'shorts out' the motor. While the motor is spinning it acts as generator, producing voltage which pushes current through SW2. The current produces torque which brakes the motor. This is dynamic braking, but not regenerative.

However if PWM is applied to SW2 then each time it opens the collapsing magnetic field in the motor's winding inductance creates a 'back-emf' voltage which tries to keep the current going. The current then takes the only path available to it, through D1 into the battery. As well as charging the battery the back-emf current also produces braking torque in the motor.

If the controller uses PWM to control motor speed then this circuit can be 'free', because it uses the same switches that are used in a half-bridge configuration. The only change required is to keep the 'motor' switch open while applying PWM to the 'brake' switch. Most controllers use MOSFETs which have built-in body diodes, so an external diode is not needed either.