Interesting idea. Doesn't "urgent care" imply that a home visit wouldn't be suitable? If I was in urgent need then a hospital visit with access to all the equipment an emergency room has would be the best option right? Seems like this kind of wording would open you up to some kind of liability issues.
Emergency care is what you are describing. Urgent care, while urgent, doesn't have the same immediacy. This replaces what gets done in an urgent care facility.
