Do I have any options for appealing my APFT result?

SSgt Joseph Baptist having graded PT tests, conducted, resourced, and ran PT tests for almost two decades, I can say with absolute certainty that in the Army more graders count reps that shouldn't be counted, than grader shave them. This is a frequent issue when Soldiers go to their first NCOES and fail because it's the first time they've been graded to standard. They swear up and down that "I've never failed XXX event" or "the grader graded me unfairly". It's so common that they now record the PT tests. When the student contests that they were not graded to standard, the record is reviewed. The overwhelming result? That the student was graded to standard. That their previous leadership failed to grade them to standard.

SSgt Joseph Baptist you're taking it at face value from a PFC that he's being kicked out of the Army. 1st sergeants like to say a lot of things better either an exaggeration or an attempt to scare someone. A first sergeant is not the chapter approving Authority. When a colonel reviews this packet they will see a soldier who was on a profile, who has less than 2 years in the army, and failed two different events. I would place a hefty wager that if this service member passed his next APFT he would not be separated. When a commander reviews a chapter packet like this they don't see a soldier who fails to perform, or cannot meet the standard. They see a leadership failure. I have worked directly for many field grade Commanders and I don't know a single one that would approve a chapter packet under these circumstances. They would definitely approve a bar to re-enlistment, but this would fail most Commanders for a chapter recommendation.
So, my advice stands that he correct his push-ups and pass the next APFT that his unit is holding.
I'm not a commander, so I happily invite any lieutenant colonel or higher for their input on whether they would recommend a chapter packet in this soldiers situation.

Remedial pt was put on halt since before block leave. I was on the bike because of a permanent profile. I have a bad back and was just granted my permanent profile on Wednesday this week. My chest was hitting the ground. I cant go any further.

Well like I said, chest hitting the ground does not make a pushup, it's the 90 angle of your elbows.
If remedial PT was on hold, then perhaps you need to supplement your own PT. PRT is not enough to build, it is designed to sustain and put the onus on the soldier to improve and build. Hence why PRT sucks.

Yeah the 101st when I was with them was strict on assigning folks to remedial PT if they could not keep up with their units PT regime. Only one item I had a bitch about was the number of people shamming with the tape test. I'm 6 foot 5 inch and each time met height and weight normal standards without a tape but I also ran after hours on my own to keep it that way. Some of these Soldiers that were on tape when I served were just FAT and eating full pizza's like twice a week. They just had the blubber in the right places to pass TAPE. I'd see that at night after hours, they would be chowing down on greasy pizza and throwing it back with beer BUT they were in shape if you asked them.