Is falsely accusing someone of immorality haram?

Haram Speech & character Quran · 2

Short answer

Yes. The Quran sets a severe punishment for those who accuse chaste people without four witnesses (24:4), and curses those who slander believing women who never thought of evil (24:23). It is a major sin.

Evidence

Quran

وَٱلَّذِينَ يَرْمُونَ ٱلْمُحْصَنَٰتِ ثُمَّ لَمْ يَأْتُوا۟ بِأَرْبَعَةِ شُهَدَآءَ فَٱجْلِدُوهُمْ ثَمَٰنِينَ جَلْدَةًۭ وَلَا تَقْبَلُوا۟ لَهُمْ شَهَٰدَةً أَبَدًۭا ۚ وَأُو۟لَٰٓئِكَ هُمُ ٱلْفَٰسِقُونَ ﴿٤﴾

And those who accuse honourable women but bring not four witnesses, scourge them (with) eighty stripes and never (afterward) accept their testimony - They indeed are evil-doers -

إِنَّ ٱلَّذِينَ يَرْمُونَ ٱلْمُحْصَنَٰتِ ٱلْغَٰفِلَٰتِ ٱلْمُؤْمِنَٰتِ لُعِنُوا۟ فِى ٱلدُّنْيَا وَٱلْءَاخِرَةِ وَلَهُمْ عَذَابٌ عَظِيمٌۭ ﴿٢٣﴾

Lo! as for those who traduce virtuous, believing women (who are) careless, cursed are they in the world and the Hereafter. Theirs will be an awful doom

This is general information, not a fatwa. For your own situation, ask a qualified scholar.