Would

"I can't say" == Nílim in ann a rá

The context is that somebody asks about something that was told to me in confidence, and I want to politely but firmly tell them that I can't talk about it.

"in ann" refers to an ability, so: Nílim in ann a rá = I don't have the ability to say it.

But this isn't meant in your context.
So: Ní féidir liom a rá./Ní thig liom a rá. = I can't say it.

Déarfainn: "Dúradh mar rún dom agus tá ceangal (dualgas) orm gan faic a rá faoi."

Nuair a deir tú rud le duine bí soiléir faoi agus ná bí ag fódóireacht timpeall ar an sceal

Nach [font=]n-insítear[/font] rúin in ionad "deir" a úsáid. Sin a déarfainn féin, ach b'fhéidir go n-oibríonn "dúradh" toisc go bhfuil "mar" in úsáid agat roimh rún?

What about, 'Níl cead agam a rá/labhairt faoi'?

The meaning would be changed to "I'm not allowed to ["I don't have permission to"] ...

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

Which is the actual meaning of the EN phrase ... hence my suggestion. In EN, 'I can't' say' has nothing to do with physical ability to say something, it is about explicit or implicit permission or prevention.