You could roll initiative
One way of resolving this is to roll initiative. This at least resembles how it would be handled in (or at the end of) combat. However, as you say, it does seem a bit excessive...
Or act out of initiative
If one of the party has spare the dying or a healing spell, or a healing potion, then this becomes trivial, since they cast the spell and then no-one is dying. Even if the dying PC rolled a 1 on their death saving throw, they're back up (or at least stable) thanks to one casting of a spell.
However, if no-one has those spells, then it's down to death saving throws and medicine checks. This therefore becomes a skill challenge rather than a combat scenario. This does not need to involve the entire party, so initiative becomes irrelevant.
Rather than having everyone dogpile the fallen PC with Medicine checks (see this question about skill checks and why it's not a great idea to have everyone repeatedly try the same skill check), it makes sense if the PC with the highest Medicine check (perhaps helped by another PC via the Help action, which gives advantage on the check) trying to stabilise the fallen PC as the PC is making death saving throws, alternating between them.
@Yakk has pointed out in comments that this makes stabilising a dying PC harder than it would if everyone were allowed a try, as is commonly ruled during combat, but I would argue that, from a narrative perspective, this doesn't make much sense. Given that rounds of combat are 6 second, stabilising someone within 6 seconds is already cutting it fine, but if each member of a party of 4 trying to revive their dying party member, what does that look like?
Are they all attempting it separately? That means either each gets 2 seconds to try before the next one has a go, which doesn't sound plausible to me, or they're all stepping over each other to simultaneously individually try to stabilise their friend, in which case I'd almost want to impose disadvantage on them for that. However, if we assume that the party are collaborating in their efforts to revive their friend, then I'd say that's what the Help action represents, and thus grants advantage on the one with the highest Medicine skill.