Framing (you can skip this)
Editorial Note: To be explicit here about what I am claiming; I am attempting to answer OP's question "[is] denying access to a paladin's aura is possible under RAW." The rules about paladin auras include the term 'friendly', and thus we need to know what this word means within the context of the larger body of rules.
I agree with KRyan that the authors of the PHB might not have had any particular rules-intent associated with this word; it might both have had only its standard English usage and had its usage shift in meaning depending on the passage. I find this entirely possible. However, I disagree with him on two points: First, that we are better off assuming there was no intent and no pattern then actually checking for one (and that the process of looking for patterns is, in itself, incorrect and wrong). And second, that if there was no intent, then there can be no meaningful application of the word 'friendly' to OP's question.
Rather, I believe that happy accidents can exist. Even if the authors did not intend to create a consistent in-game definition of friendly, they could have done so unintentionally, and that definition could still be useful to OP and their DM. If the word is used consistently throughout the corpus in ways that connect logically to other powers and abilities, that may be useful in their game, regardless of intent.
Actual answer starts here
What does friendly mean?
There is no game definition of what is a friend. However, the PHB does tell us what being "friendly" means (p. 185):
In general terms, an NPC’s attitude toward you is described as friendly, indifferent, or hostile. Friendly NPCs are predisposed to help you, and hostile ones are inclined to get in your way
And the DMG adds to this understanding (p. 244):
A friendly creature wants to help the adventurers and wishes for them to succeed. For tasks or actions that require no particular risk, effort, or cost, friendly creatures usually help without question. If an element of personal risk is involved, a successful Charisma check might be required to convince a friendly creature to take that risk.
So by definition, a friendly creature wants you to succeed, but is not always willing to risk itself to help you. We can also note that about a third of the time when the PHB mentions allies, it also mentions them as being friendly. The two terms seem to be related1:
PHB, Help Action, emphases mine:
Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally's attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.
PHB, Forest Gnome (p. 37), emphases mine:
Forest gnomes tend to be friendly with other good-spirited woodland folk, and they regard elves and good fey as their most important allies. These gnomes also befriend small forest animals and rely on them for information about threats that might prowl their lands.
It is clear that being an ally is linked to being friendly, and it appears that allies are always friendly2.
What is the difference between an ally and a friend as they relate to other rules? Here we can return to the DMG definition of friendly, and note that while a friendly creature always wants its friends to succeed, it is not always willing to risk itself to help. While a friendly creature wants you to succeed, an ally is a friendly creature that is furthermore willing to incur risk, effort, or cost to itself in order to help you succeed. That an ally is willing to incur personal risk on your behalf is consistent with all the uses of 'ally' in the PHB. It is especially congruent with the definition of the Help action, where you must place yourself within five feet of a foe to give your ally advantage3. And it even touches on the distinction made in the description of Forest Gnomes, where small woodland creatures are just friends, but more combat-ready fey and elves are not only friends but allies as well.
Thus, any ally is always friendly, but not every friend is an ally. A "friend" of the paladin is someone who is predisposed to help them, but this is a less stringent definition than an ally - they may be just 'fair-weather friends' who welcome the beneficial auras the paladins provide without being willing to risk themselves on the paladin's behalf, or inconvenience themselves by attempting to follow the paladin's code, or moderate their behavior to retain the paladin's favor.
Only at the point in time in which the paladin's companions cease to be predisposed to help the paladin, and become indifferent to the paladin's goals, would they cease to be friendly and would thus no longer receive the benefit of the paladin's aura.
The paladin does not actively choose
Typically abilities that a character elects to use have "can" or "choose" in their language. Your paladin's Aura of Protection says:
Starting at 6th level, whenever you or a friendly creature within 10 feet of you must make a saving throw, the creature gains a bonus to the saving throw equal to your Charisma modifier (with a minimum bonus of +1).
"The creature gains a bonus" leaves no room for the paladin to decide. If the paladin could extend this benefit or not, it would say something like "whenever you or a friendly creature within 10 feet of you must make a saving throw, the creature can gain a bonus to the saving throw". Similarly, your Aura of Courage says,
Starting at 10th level, you and friendly creatures within 10 feet of you can’t be frightened while you are conscious.
It doesn't say "you and friendly creatures you choose can't be frightened".
Within the description of the abilities, the paladin cannot choose anything - they will provide the benefit to those who are friendly to the paladin, regardless of the paladin's feelings or intentions.
A caveat - the paladin's abilities are powered by their Oath
A paladin may or may not worship a god, but even if they do, that god is not what is powering the paladin's abilities. The source of a Paladin's power is their Oath - their deep personal commitment to their values. Like their spells, their auras are a function of the paladin's belief (and they don't work while the paladin is unconscious).
Even though the paladin doesn't have a choice about to whom they extend the benefits of their aura, as a DM, I would look hard at the tenets of the paladin's Oath, and in some cases, let their guiding philosophy decide. A Paladin of Mercy believes in:
Forgiveness. Let bygones be bygones. Forgive those who trespass against you.
Redemption. Those who aren't evil can be redeemed. Show them the error of their ways and ensure they are on the path to redemption.
Here, you are stuck with your jackass friends. In fact, the behavior you are willing to suffer on their behalf is what is powering your abilities to begin with. I wouldn't think a Mercy paladin would be able to deny their aura to anyone who legitimately sought it.
Contrast that with the Oath of Conquest:
Rule with an Iron Fist. Once you have conquered, tolerate no dissent. Your word is law. Those who obey it shall be favored. Those who defy it shall be punished as an example to all who might follow.
Strength Above All. You shall rule until a stronger one arises.
As a DM, if a player's Conquest paladin wanted to deny their aura to 'friends' until those friends straightened up and flew right, I would permit it - not because the paladin wanted to, but because their belief required it.
1 Similar links between 'friendly' and 'allied' can be found in the descriptions of Character Building (Building Bruenor, Step 4), the Tiefling Race details, and the Bard Class Feature, Song of Rest.
2 There are places in the PHB where being an ally is mentioned without an explicit mention of friendliness. We can look to these to see whether it is ever possible to be an ally without being friendly. A review of who a Bard inspires (p.54, in two places), the functions of the Cleric class (p.56), who can benefit from a Battlemaster's Distracting Strike (p.84), who a Ranger can teach (p.90), who can benefit from a Wizard's Illusory Reality (p.118), who a flawed Folk Hero has trouble trusting (p.132), who might be healed with the Healer feat (p.167), and who can be targeted with an Aid spell (p.211) finds that in no case is it suggested that an ally would not be friendly.
3 It also makes sense with the feature Pack Tactics (found in the PHB descriptions of Dire Wolves, Lions, Reef Sharks, and Wolves), which confers advantage on your attacks when an ally of yours is willing to risk themselves by being adjacent to your shared foe.