No, it's not possible to completely correct for LED lights, parts of the colour spectrum are missing. Colour temperature meters cannot read them either. People do use them of course, and you can "sort of" correct, with gels and post, but subjects will never be accurately reproduced in the same way as can be done using continuous spectrum lights such as tungsten or HMI.
It depends upon the kind of intended use and subjects you shoot of course; and what you might find "acceptable" as far as colour balance and reproduction are concerned. I do not do it, but for run and gun kind of video work, and if I needed an on camera light, I would choose the Lowel i-Light or dimmable id-light.
If you run tests, shooting the same subject - say a person in blue cothing, first using corrected tungsten light, then use LED's, you'll see a marked difference in the way the subject is reproduced. The latter will be devoid of the subtle shades of blue, skin will look "flat" and lack colour. It's probable most people do not see the comparison side by side; but if you do, it's then that the LED deficiencies really show up.
Peter
