This really comes down to whether the hospital considers you an inpatient or outpatient case.
Inpatient means you are admitted into the hospital per doctor's order. Staying in a hospital overnight does not necessarily signify this. If you aren't formally admitted, and are considered to be under observation, you could be considered "outpatient" even if you stay in the hospital overnight.
When you are inpatient, Part A covers your hospital services for 60 days after you pay a deductible.
When you were outpatient, Part A does not cover any services. These would be covered under Part B (and a supplemental plan) and you would be responsible for any copayment after a small deductible is met.
Medicare will not cover follow up care in a nursing home for outpatients. You must spend 3 consecutive nights as an inpatient in order to be eligible for nursing home coverage.
So, what does this mean for you? It means that even when you are dealing with a major medical issue that requires hospitalization & nursing home care, you need to be aware of what type of patient the hospital considers you (or have someone with you that can be making sure they know). The question to ask the hospital is whether you are considered inpatient or outpatient. Either way you will need 3 consecutive nights as inpatient to have nursing home covered.