I have a roommate who does not have a lease (we are in PA). We had an argument so of course now she told me she's moving out. She just told me June 3rd, but thinks she can move out in a week and just pay me for that week (because she hasn't paid for June at all yet). If she's been living with me for over a year paying me monthly and I'd have to give her notice to move out, wouldn't she have to give me more notice and be expected to pay for the whole month?

Yes, she has to give you 30 days WRITTEN notice of her intention to vacate. If she does not give such notice, then she owes you for 30 days from when she gives you that notice or vacates.

If she gave you a deposit, you can hold that deposit against unpaid rent.