Ross had previously named Dowie as one of the women with whom he had an extra-marital relationship while National MP for Botany.
The text message included the words: "You deserve to die."
Ross initially received the message in August last year, but has claimed that he read it two months later as he was considering self-harm.
He was taken into mental health care shortly after.
Ross revealed the police investigation during a television interview, which was apparently a call to the Crimestoppers hotline. Ross said he did not lay the complaint.
A spokesperson for the Police said: "After an anonymous call to Crimestoppers, Police commenced a thorough investigation into this matter. The investigation has determined the text communication sent does not amount to sufficient evidence of an offence under the Harmful Digital Communications Act and as such no further action will be taken. "
WHERE TO GET HELP:
If you are worried about your or someone else's mental health, the best place to get help is your GP or local mental health provider. However, if you or someone else is in danger or endangering others, call police immediately on 111.
OR IF YOU NEED TO TALK TO SOMEONE ELSE:
• 0800 543 354 (0800 LIFELINE) or free text 4357 (HELP) (available 24/7)
• https://www.lifeline.org.nz/services/suicide-crisis-helpline
• YOUTHLINE: 0800 376 633
• NEED TO TALK? Free call or text 1737 (available 24/7)
• KIDSLINE: 0800 543 754 (available 24/7)
• WHATSUP: 0800 942 8787 (1pm to 11pm)
• DEPRESSION HELPLINE: 0800 111 757 or TEXT 4202