How do you stop this from happening when trying to slide two or more strings at one time?

I'm finally trying to learn the song Johnny B Goode right now and the slide from 4th to 5th fret on the highest two strings at the beginning is cutting my index finger to ribbons every time I practice.

The tip of my finger is calloused but not on the part that the high e string touches, but sliding with just the tip on the 2 strings doesn't feel right to me at all.
Doesn't look like you have much callous to protect your fingers. Not played regularly for awhile?

I'd recommend putting a BandAid on that finger before resuming practice. I'd also recommend some thin gloves for your fretting hand in the short term, if you want to continue practicing before the finger heals. It's awkward, but practice can still be useful with a fretting glove.
Hmmm that's odd. But try getting a bottle of new skin or even super glue and put it on before you play.

Ps which part are you playing with 4th to 5th frets?
This has happened to me after doing excessive slide practice that my fingers aren't accustomed to. I would personally recommend giving it a rest for a day or two then getting back to it.
Only time I've ever cut myself while playing was while playing bass, only like 2 or 3 months after picking it up for the first time. I had developed a very big and nasty blister on my pointer finger, but my buddies wanted to jam, and I decided to play through the pain, because we were teenagers trying to be a metal band, and that was badass. During a totally sweet breakdown (I promise, it was awesome), I did a big ol bass drop slide, on the low string, down from the 12th fret to open. Busted that blister right open, and spilled blood all over the fretboard. Took a long time for the oozing shaped blood stain in the rosewood to fade away. Metal as shit. \m/
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focus on using the tip of the fingers, if you don't want to do that then leave that song alone for a couple of days and take care to not worsen the cut when you are playing and eventually that area of the finger should become strong enough to play that way, though i would definitely recommend using the tips.
1) you're pressing way too hard

2) you're probably pressing too hard because you're putting the pad of your finger on the double-stop

When doing double stops it's OK to roll your index finger slightly to the outside. It's really the only way to do a double stop and still use the other fingers.

If you actually have to press that hard to get sound, your guitar probably needs adjustment. The action (distance between strings and fretboard) could be too high. Old corroded strings also impede slides.

I'd say practice the part slowly and use as little pressure as you can to get the sound.