Bottom line, I want to make it easier for my level 4 fighter companion to heal other PCs in combat or stabilize them if they reach 0 HP. How should I go about that?
This part is easy: give them a Healer's Kit. That lets anyone stabilize another character without having to make a check, and it only costs 5gp.
Beyond that, if you want to multi-class to get as much healing as possible, your best bet is to go Cleric with the Life domain. The Cleric has full spell-casting slots and the largest number of supporting and healing spells, and the Life Domain makes your healing spells even better.
You need do nothing.
Having a healer is nice, but it's far less necessary in 5e than it used to be. It's actually pretty difficult for characters to die in this version. A few potions will be a good idea. A few Healing Kits for stabilizing anyone who goes to zero. Maybe someone takes the Healer feat. Maybe someone takes the Magic Initiate feat, and goes with cleric. Or even someone dips a level in Life Cleric.
Few battles last long enough for people to die in, even if they get dropped to zero hp. So if the party is vigilant to keep an eye on anyone that goes down and to stabilize them before they fail the 3 death saves, they can then take rests as needed to heal back up. According to the rules, a short rest can get you back most of your hp and a few of your resources, and a long rest all of your hp, spells, etc.
So, to directly answer your actual question: I would do nothing to "make the battle encounters balanced with no healer"; it isn't necessary.
Come to think of it, I don't think I've played with a 5e campaign yet that had a serious or dedicated healer.
In the campaigns I've played much in, one went through Out of the Abyss with a 3-PC party of a fighter/rogue/warlock multiclass, a battlemaster fighter, and a vengeance paladin. The other campaign had a large party in a homebrew world, and little healing: champion fighter, eldritch knight fighter, druid, assassin rogue, shadow monk, wild magic sorcerer, beastmaster ranger, wizard.
The campaigns I've run for, one is a smaller party, of a barbarian, battlemaster fighter, wild magic sorcerer, vengeance paladin, and sometimes an NPC valor bard. The other is again a large party, but little healing: champion fighter, thief rogue, vengeance paladin, valor bard, moon druid, warlock, barbarian, fighter/knowledge cleric multiclass.
It looks so far to me like 5e can work pretty well with almost any combination of classes and races, so I'm inclined to not worry about that, and let the players play whatever they want to.
Best Answer
It won't break the game, but you should not do it
Guaranteed healing is not fun, and the players will probably abuse this mechanic instead of finding clever ideas of avoiding damage. There are report of parties here with no healer at all, and most of these parties found other ways of neglecting damage, like using traps or crowd-control, carefully planning attacks, using diplomacy, etc.
Try playing normally first, and if your party really struggles with surviving, you can lower the CR of encounters a little, or provide them with some healing potions. A player could also go for multiclassing if the party thinks it can help, no problem with that on a long campaign!
There is a saying in software development which says "Premature optimization is the root of all evil", and I think it applies here as well. Be careful with what you give to your party, because you might not be able to take it away later.