My sister texted her love this:

ne yere ne göğe ismini yazdım
senin ismini aşkım kalbime
YAZDIM...

Since then, he hasn't replied. Could it have offended him? She thinks it means "I wrote your name not on the skies nor on the stars,I wrote your name on my heart"

They are apparently in love, but could this text have offended him from a Turkish perspective?

The text is all right. Nothing wrong with the text.