Is it legal if my employer harasses me or tells co-workers about my situation?
No. Your employer cannot fire you, harass you, put you at a lower-ranked position, or punish you in any way, for asking for this time off or for taking this time off.1
Also, the law requires your employer to keep your domestic violence situation confidential.2 For example, your employer cannot tell your co-workers, your clients, or other employers that you took time off to deal with domestic violence. Your employer also cannot disclose (cannot talk about or write about) the reason for your time off. S/he must also keep private any documents that you give him/her that relate to your domestic violence situation.
1 Fla. Stat. § 741.313(5)(b)
2 Fla. Stat. § 741.313(4)(c)