This is my most practiced mu; my brother is a doc main.
Overall, I think Doc is an incredibly underrated character, for the simple reason that Mario is better than him. You you want to have an easier time, use Mario. Because of this, Doc is generally thought of as a bad character, when in reality, he's very good, but just worse than Mario.
In short, I think the mu is either even, or 60:40 Doc's favor. Doc has an easier time in neutral, has better combos, and great kill options. Our ability to force him offstage and edgeguard brings us back into the mu, though.
To go into more detail:
1. Doc's UpB is dumb. Like, really, really dumb. It's a 3 frame kill move that can be used in the air and has a huge hitbox. If you jab him with any sort of spacing and don't have sufficient rage, he can break your rapid jab (or jab cancel) with this move, usually doing 26% and sending you offstage. I've lost many a winning position from hitting with jab, being sent offstage with UpB, and being edgeugarded. It's pretty disgusting. Jab has to be used up close, or with rage. Otherwise, spacing with jab is incredibly hard here. Which is a shame, because our jab is normally great.
2. Doc wins in neutral. His projectile is great, he can throw it from a full hop or shorthop from a platform, and follow up, allowing an approach. Nair will beat this inconsistently; you have to carefully time a nair hitbox to beat the pil and continue towards doc; uair is normally better. Doc's bair hits hard oos and can never (NEVER) be punished if he chooses to retreat while using it. This actually gives Doc another advantage in the mu over Mario, who can be punished for throwing out bairs without thinking. Uair can only be punished if he lands with it. Jabs clank and are the same speed, he can use dtilt to go under your jab, though. Utilt combos absurdly well at low percent and is very hard to punish on shield, although it can be punished on powershield. He also has large dash grab range that can grab you at the peak of your short hop. All these things make life hard for us. Falco should be using lots and lots of dtilt, which is safe on shield, and almost safe on whiff. Jab is generally unsafe for reasons mentioned earlier. Falco should approach with dash --> tomahawk mostly, as spaced retreating fair/bair are completely safe. Lasers help break him momentum, but his projectile and reflector are better than ours, so use them sparingly. Doc's usmash is almost never punishable on shield/whiff (it has to be the worst possible spacing by him), only the front hit of dsmash is punishable, and his fsmash is only punishable on shield if misspaced. His fsmash also retreats before the hitbox comes out, making it an incredibly safe punish.
3. Falco has good follow ups, but has to be careful. Due to Doc's floatiness, fast aerials and aerial mobility, we don't combo him well, so you need to be sure of your confirms. Overshooting normally leads to a heavy punish in his favor, as Falco is combo food. Dtilt combos well into grab and fair at low percents (I recommend fair in most scenarios over nair, as nair trades with Doc's nair [3 frame aerial], which fair beats it from the front). Use this to try and get him offstage. Dthrow to dash attack or fair is best until about 40% imo, as these allow for the best followups. Dthrow to usmash allows him to get away, so it's worse in the long run. Uthrow should be used after 40% as it will lead to one uair (rarely more), unless he's near the edge, in which case you should throw him off. Uair frame traps well, but chasing is hard as his aerial mobilioty is faster than our ground mobility. We can pressure and often force an airdodge, but he gets away better than most. If Doc grabs you around 70% and lacks rage, he can true combo you into a fair, regardless of DI. At higher percents (90%ish) you can airdodge his fair, but get frame trapped into a dsmash unless you DI offstage. If Doc has too much rage, or your damage is too high, you can jump away.
4. Doc wins the air game. He moves in and out much better than us, his bair/uair cleanly beat anything but bair (which trades) and have almost no lag. His nair trades with ours, and can beat fair from some angles. He doesn't have a great answer to uair, but can airdodge through and away. Our bair will miss him oos if he's dashing or landing with an aerial; he can kill us for this. His uair strings together ridiculously well, so you really don't want to get caught out by one of those. His bair isn't safe on hit at zero, we can punish with a dash attack on hit. Also worth mentioning that it is virtually impossible to punish Doc's dash attack with a bair. His profile during the attack is so close to the ground, I feel the sweetspot punish is frame perfect in this scenario. Really hard.
5. Both characters can edgeguard each other really well. If falco is offstage, and Doc is standing near the ledge, falco becomes unable to use SideB as it may get reflected, essentially ending falco's stock. Instead, using Side B above/below the ledge and recovering with jump + upB is often preferred. Due to this threat, Falco will be recovering low really often in this MU, you need to be really, really proficient at teching. Tech --> wall jump --> phantasm onstage should get you onstage before Doc in almost every scenario, giving you stage control. Teching can be difficult though, Doc's sweetspot on his upB and bair has a very different tech timing to his sourspot on those moves, so you have to react to the hitstun; this is hard but doable. He can spike you if he spaces his tornado properly, but this is only feasible if falco is forced to recover directly vertical (on a stage like lylat). Doc has lots of trouble recovering too, however, fair significantly limits his recover options. It covers airdodges well, and the front hitbox will trade with upB (once again, you need to tech) and will spike him. If he gets horizontally close to the ledge, such that his upB can challenge fair from below (upB will win), you should dair instead; the sourspot will kill anyway.
I think that's most of what I wanted to say. Doc is much better than he's given credit for; if we couldn't edgeguard him, we'd lose this mu real bad.
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