There are three types of fire in Game of Thrones and in order of least to most powerful they are:
- Fire
- Wildfire
- Dragon fire
From those three, we've only seen the White Walkers interact with one of the types of fire: fire. Here the fire parted around them to let them pass, how this happens is speculation but probably because "They bring the cold with them". See the below image from Season 6 Episode 5, "The Door".

I believe a similar image is shown in Season 7 Episode 6, "Beyond the Wall", when the Night King and some White Walkers have fire around them.
On the other end of the spectrum we have dragon fire, this is speculated to be able to kill White Walkers but as of yet we have not seen that happen. It is speculated because of dragon glass, or obsidian, which has been known to kill them, see my related answer here.
Lastly, we have wildfire which I don't think has been seen to interact with White Walkers and even their wights. It is a mix between the two though probably closer to normal fire than dragon fire. I would speculate that it would not kill them but that is just speculation on my part and we are likely to know more in Season 8
now that the White Walkers have broken through the Wall and man is going to throw everything they have at them.
On a side note the following statement of your is incorrect, we have only seen fire, and dragon fire for that matter, kill wights but not White Walkers.
White Walkers can be eliminated by using fire.