There was no one lined up to succeed Eärnur.
The issue wasn't really the manner of Eärnur's death, but the fact that there were was no one around with a strong claim to the throne afterwards, and that Gondor had recently gone through a civil war and did not want to risk another one by accepting someone with a weak claim
Now the descendants of the kings had become few. Their numbers had been greatly diminished in the Kin-strife; whereas since that time the kings had become jealous and watchful of those near akin. Often those on whom suspicion fell had fled to Umbar and there joined the rebels; while others had renounced their lineage and taken wives not of Númenórean blood.
So it was that no claimant to the crown could be found who was of pure blood, or whose claim all would allow; and all feared the memory of the Kin-strife, knowing that if any such dissension arose again, then Gondor would perish. Therefore, though the years lengthened, the Steward continued to rule Gondor, and the crown of Elendil lay in the lap of King Eärnil in the Houses of the Dead, where Eärnur had left it.
The Lord of the Rings - Appendix A I (iv) "Gondor and the heirs of Anárion"
So even if the Witch King didn't kill him, but he died a natural death several years later, the situation would be the same as the situation present in The Lord of the Rings.
However, possibly had Eärnur not been killed by the Witch King, he'd have eventually produced, adopted, or appointed an heir.
He was already 122 at the time of death, which means he probably had another couple of decades to live, judging by the lifespans of the previous couple of Kings.
Possibly his "hot mood" would settled down and/or he'd succumb to pressure from his advisors for the need for an heir.