Started up my car (manual transmission) and when I released the clutch pedal it won't push back down anymore

Well, it would shift easily with engine off, because the transmission input shaft is not turning. Engine running and clutch engaging good, the input shaft would again be stationary and allow smooth shifting.
What you have to determine is if the clutch disc or pressure plate is bad (a new clutch is needed), or if the pedal hydraulic system has a malfunction. When you push the pedal down, a rod from the top of pedal pushes into the clutch master cylinder bolted to the firewall under the hood. This cylinder has a reservoir filled with brake fluid. When the rod forces fluid from the clutch master down a steel line and into the clutch slave cylinder down on the transmission bell housing, the hydraulic pressure forces a plunger from the slave cylinder to extend and push the clutch fork lever a small amount, enough to force the clutch release bearing (aka throw-out bearing) against the pressure plate. This relieves pressure on the clutch disc and input shaft becomes stationary for easy shifting.

The axle job should not have caused any problem to develop in the clutch.
I don't see how they could have.
If you can work the slave by hand, it may be a problem with the clutch master cylinder. Maybe the rod from clutch pedal into the master is bent or binding up and won't travel into the master. You could check if master operates easily by taking line off at the slave cylinder (or if not able to remove hose, then open the bleeder valve on slave) and see if your pedal operates now. Use a can to catch fluid if you don't want it to spill on floor. If the pedal still won't move it must be a problem with the master cylinder or the clutch rod.
If the pedal works with line removed, or the bleed valve open, then the release bearing inside the bellhousing must be binding up and not moving forward to engage clutch. (The slave cylinder pushes the clutch fork lever which pushes the release bearing forward on a shaft up against the clutch pressure plate to allow the clutch to operate .If the release bearing is the problem, the transaxle will have to be removed to fix it.
As I said, though, if you can operate the slave and the cllutch lever it pushes with your hand, it doesn't sound like it is binding down there. Check the pedal's push rod going into the master cylinder. Good luck.

Your clutch is not releasing, check the slave cylinder. Consider bringing it to a professional.

That cylinder is known as the clutch slave cylinder. It should push against the clutch fork every time the clutch pedal is depressed to allow gear changes. If it doesn't, suspect a faulty slave cylinder, or clutch master cylinder.

here's what you can do to fix your clutch.

1. if your clutch still uses clutch cable (there is a cable between the transmision and the clutch pedal), the you can adjust the clutch release by tightening/loosen the clutch pedal. adjustments can be done on the transmission side of the clutch cable.
2. hydrovac clutch with self correcting feature. just pull your clutch pedal for a few seconds and check if the clutch release changes.
3. hydrovac clutch without self correcting. between the clutch pedal and the clutch master, you can see a piece of metal rod that connects the 2. this metal is called push rod. there are screws/nuts at the clutch pedal to adjust this push rod. just adjust this screw to your prefered clutch release.

It's called the clutch switch and it's supposed to only allow current to the starter when the clutch pedal is depressed all the way down, releasing the clutch. It's bolted in front of the clutch pedal. Yours must be bad or shifted to the left somehow. Have it checked, then repositioned or replaced.

do you mean it wont turn over until you push the clutch pedal in, then after several attempts it finally did turnover and run???? clutch switch may be going bad.

the mechanic should place or connect the clutch pedal spring to the clutch pedal .connect the push rod to the clutch pedal.then connect the push rod to the clutch master cylinder then refill the clutch master cylinder .by means of pressure tube connect the clutch slave cylinder .then connect the clutch slave cylinder to the push rod .And the push rod connect it to the clutch pork. the clutch pork connect to the spring. then connect the fly wheel , the clutch linning ,the clutch disk ,the release bearing,then the clutch pork to the transmission shaft . then connect the transmission assembly.And you can goon with your date.

The problem is probably not the throwout bearing. Noise from throwout bearings act just the opposite, you only hear them when you push in the clutch pedal. Noise when the clutch pedal is released only is either a bad pilot bearing (probably oilite bushing) (the bearing/bushing in the middle of the flywheel that supports the transmission input shaft), or bad transmission bearings (will happen sometimes but is unlikely because in neutral, with the clutch pedal released, tranny bearings spin, but are not really under any load, but in some cases they might be the cause of the noise anyway)

you have civic SI type R with the K20A engine and 6sp transmission? correct?

yes the clutch is adjustable.

1. loosen the clutch pedal position switch(lower of the 2 switches) locknut and back off the clutch pedal position switch.
2. loosen the clutch push-rod locknut and turn the push-rod in or out to get the desired height and stroke of the pedal.
3. tighten the push-rod locknut
4. with the clutch pedal released, turn the clutch pedal position switch in until it contacts the clutch pedal.
5. turn the clutch pedal position switch in an additional 3/4 to 1 turn.
6. tighten the clutch pedal position switch locknut.
7. loosen the clutch interlock switch(the upper switch) locknut and then the interlock switch.
   
8. press the clutch pedal to the floor.
9. release the pedal about 1/2 in. from the fully depressed position, and hold it there. adjust the position of the interlock switch so that the engine will start with the clutch pedal in this position.
10. tighten the interlock switch locknut.

clutch pedal stroke should be: 5.1 - 5.5 in
clutch pedal height should be: 7.76 in

if adjusting the clutch doesn't help with the transmission feeling notchy, may be the syncros.