Is what the girl did—ending her relationship with a young man she loved online because this relationship is not pleasing to Allah—correct? And would she be sinful if she checked on him from time to time?

Familiarity, love affairs, and infatuation between young men and women, even if for the purpose of marriage, open a door to fitna (temptation/strife) and corruption. One must be cautious not to be lax in these relationships. Cutting off the relationship with the young man is the correct action, and you must remain firm on it and not speak to him under the pretext of reassurance, for this is one of Satan's stratagems. If the young man is capable of marriage, let him propose to the girl's family and seek her hand. Otherwise, she should divert her heart from him and occupy herself with what benefits her.