If your saw requires 220 VAC, you MIGHT still be able to make it work. It all depends on how power is supplied to your garage from the house panel.
The current garage outlets are labeled "L5-20", so they probably are 120 VAC, 20 amps. Now, are those fed directly from one cable from the house? Or, is there a fuse or breaker box in the garage fed from the house, with your two outlets run from that box? You need to examine closely the cable from the house breaker box out to the garage. If it was done really simply, it will have only two current-carrying conductors - one black (Hot) and one white (Neutral), plus a bare copper Bonding or Ground wire. In that case, you really should be planning to replace that cable. But if it has three current-carrying conductors (those above, plus a red (second Hot) we're getting closer. In that case there is a good chance the black and red are connected to opposite sides of the 110 / 220 VAC bus system in your house breaker box, via two terminals on a duplex breaker. If so, you will find that there is 220 VAC between black and red, and that each of those two is 110 VAC from the white (Neutral) line. If that is true, there is still one more thing to consider.
I would suspect your new saw has a big motor and that's why it requires 220 volts. But that also may mean that the current involved is lower than if it had been 110 VAC. So look closely at the motor name plate or in the saw manual and find out what is the MAXIUMUM (starting) current required. That will dictate the size of the wiring needed to feed it. You really should consult en electrician for this next bit, but I'll outline a couple of "rules of thumb". 14-gauge wire can handle 15 amp load, 12-gauge can handle 20 amps. BUT for longer cable runs (like from the house to the garage) feeding larger inductive loads you usually go at least one gauge size larger, so figure at least 12-gauge for a 15-amp MAX starting current, and 10-gauge for 20-amp. In fact, since the saw manual recommends an L6-20 connector system, assume you need to provide a solid 20-amp supply and that will need at least 10 gauge wiring for the long run.
IF you have a cable marked "10/3", meaning 10-gauge wire with 3 current-carrying conductors, feeding the garage from a duplex breaker rated at least at 20 amps, you already have the minimum necessary to wire up your saw in the garage. In that case at the garage end you could disconnect both your existing outlets and create a new outlet for the saw with a NEMA L6-20 socket plus, of course, a matching plug on the end of the saw motor's cable. You must make sure to wire the socket and plug correctly. I would guess it is done with the black and red lines connected to the two NON-Ground blades and no connection to white, because the saw does not use Neutral for 110 VAC. The outlet should be clearly labeled "220 VAC only" to ensure nobody tries to plug in a low-voltage device with a L6 plug on its cord.
Now, that gets all the power you need to the saw, but you have no other outlets left. Oh, what about lights - where are they powered? Technically you might (depends on code requirements for an inductive motor load on the circuit) be allowed to connect other 15-amp outlets and / or lights to the lines to use 110 VAC (either black to white, or red to white), but you risk overloading the circuit by using the saw at the same time as other tools. That would blow the 20-amp breaker back in the house panel and you'd have to go there to reset it. But at least that would be electrically safe, if inconvenient. The much better solution is to have much heavier wiring supplying the garage in the first place. The proper way is to have a heavier cable, like 6/3 or something, fed from a properly-rated breaker in the house panel, feeding out to the garage and into another breaker box there. In that small box you can create a few separate branch circuits - one 220 VAC 20-amp circuit for the saw using 12/2 cable (this is a shorter cable run, and does not need Neutral), plus a couple 110 VAC 15-amp circuits for outlets and lights. If you are going this route, DEFINITELY consult a licensed electrician to get all the cable sizes and types, and all the breakers and boxes, designed correctly.