Is this sexual hsrrasment or slander

I (female) work at a restuarant as a food server. I brushed up against a fellow co-worker (male) and noticed he had very soft skin. So
I commented on it and asked him what kind of lotion he uses because I have very dry skin. He said he doesn't use lotion but gave me the name of the soap he uses. Another co-worker was standing there and I asked her to feel his arm to confirm he had soft skin. My manager walked by about that time and gave me a odd look. Next thing I know I'm taken in the office to sign a write up for sexual harrasment. This write up is then put in my employment file against me I guess. I dont want this in my file and I know I have have wrongfully be accused. What are my rights snd can that file harm me later in life. please help.

What did this "write up" state? Did you sign it?

Former employers are often reluctant to share that kind of information out of fear of lawsuits. Usually, they'll just verify your title and the dates of your employment.