Is it “proper” to give my job a one week notice while I’m on PTO?

I really want to quit my job. I love what I do but I work in such a toxic work place (I’m a caregiver). There’s a coworker of mine and she constantly bullies me and harasses me (well call her Lucy) She has spread rumors about me that I’m a homewrecker, a sl*t and a wh*ore, had other coworkers believe it too and I was bullied around the workplace. I went to my supervisor about this (we’ll call him idiot) and he just told me that “it sounds like a personal problem so he’ll stay out of it” I worked 3 weeks no weekends pulling doubles back to back training temps because every other coworker of mine was out due to covid. There’s a medtech promotion that I’ve been asking for since October of 2020 and he’s given it to 3 other people who’ve quit right after training. A few days ago he asked me to come into his office and asked me if I want the open medtech position, I said “I would love the position sir. I’ve been waiting for this opportunity and I thank you for giving it to me” then last night I find out he’s giving it to Lucy, the girl who bullies me, the girl I’ve told him about.
I just dont feel appreciated in my work place. I do everything correctly, I go above and beyond, I don’t abuse my residents I take care of them. I’ve been complimented by people above my boss and I’ve been noticed my corporate.
im on PTO right now and I wouldn’t be back until next week. After finding out my boss pulled the medtech promotion from under me and gave it to Lucy really upset me, especially after everything I’ve done for the company. I don’t want to go back. I have an offer from a different company and I’m ready to take it
Is it “proper” to give my job a one week notice while I’m on PTO?
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