Q:

Does running LED at higher voltage damages the LED?

I want to run 12v 6 watt LED at 19v by keeping maximum watts constant (current controlled). By doing so will damage the LED?

I.e
12v x 0.5000 A = 6W
19v x 0.3157 A = 6W

update1: I am running these LED with buck based PT4115 home made led driver

A:

The idea sounds good but won't work. LEDs cannot magically adjust their incoming current to suit the prevailing applied voltage unless it has some type of switching regulator built into it.
On the understanding that it doesn't, if you increase the voltage a little bit across a partially conducting LED, it will draw a lot more current and quite possibly enough to near-instantly fry it.
Take the example of a 24 ohm resistor. With 12 V applied it takes 0.5 A but with 19 V applied it takes 0.792 A. There is no power regulation implied in ohms law.

A:

If you were able to simultaneously control voltage and current across a device, all the Current-Voltage Characteristics (including the Ohm law) would be nonsense.
If you regulate in current the voltage will step down, if you regulate in voltage the current will step up, but you will always be on this curve:

