That would be kuraci :)
(Though perhaps "care for, treat" is more fitting as according to PIV, a bettering of the state of the patient is not a guaranteed part of the meaning - it uses provi "try to" in this connection.)
If you speak French or Spanish, then courir or correr "run" is the cognate of kuri.
Compare also a courier (who runs quickly to deliver important mail) or a corridor (which runs along the length of a building).
A kuracisto is someone who heals, a doktoro has a doctoral title.
A PhD in economics is a doktoro but not a kuracisto, while a village healer can be a kuracisto even if he never went to university to become a doktoro.
(Though PIV says that a kuracisto has "successfully studied medicine" and is therefore "permitted to heal people", so my last part may not be accurate.)