Why couldn't my aircraft depart fully fueled on a 44C day?
There are two problems:

First, at higher temperatures, the air is less dense; therefore there is less oxygen (by mass) in every cubic metre of it; therefore more air must be ingested by the engine (by volume) for the same quantity of fuel to be completely burnt. If the intake flow rate of air is fixed, then less fuel can be burnt and less power developed compared to a colder air temperature.

The second problem is that the lower density of air reduces the lift generated by the aircraft's wings at any speed. To make up for this the aircraft can travel faster.

In practise this means that aircraft operating in "hot and high" conditions—altitude also affects air density—will require longer runways to take off.

If it gets too hot then the aircraft might not have enough runway to get airborne.

For safety reasons the runway must be long enough for the aircraft to come to a stop if the commander decides to reject the take off at the last moment. But fast, heavy aircraft take a long time to stop; so this means the runway must be even longer.

If the aircraft is lightly loaded then it is not such a problem; there is less mass to accelerate and a lower speed must be obtained for take off. Thus a short flight simply takes a bit longer but still gets off the ground. But for AUH-SFO, this is a very long flight and will require a great deal of fuel to be taken onboard (at a guess I would imagine something on the very rough order of 80 tons). The specifics will vary by aircraft, and by how much is on them.

https://en.wikipedia.org/wiki/Hot_and_high