Sofia Ashraf, 29, posted a Facebook live video of her singing a song, My Vote Is Not For You, in front of the new chief minister's residence.
Sasikala Natarajan, a close aide to the late chief minister J Jayalalitha, was appointed to the post on Sunday.
The decision has fuelled anger as Ms Natarajan has never held public office.
She was made general secretary of the AIADMK party after Jayalalitha's death in December.
The recording of Ms Ashhraf's live video has been viewed 45,000 times and shared almost 1,000 times since it was uploaded to Facebook 18 hours ago.
Tamil Nadu to get second woman leader
The Indian rapper who took on a multinational
"The youth are getting increasingly agitated, and this appointment of a woman who has never stood for election being made chief minister was the straw that broke the camel's back," Ms Ashraf told the BBC.
"It has made us lose faith in democracy."
The AIADMK had earlier explained its choice of chief minister.
"Like Amma [mother], we wanted a strong woman leader to serve the people," party spokesperson CR Saraswathi told the AFP news agency, referring to Jayalalitha by her popular nickname.
Ms Ashraf says she was performing at a venue two streets away from Ms Natarajan's residence in Chennai (Madras) when her band decided to change the lyrics  to one of her songs.
She and others marched towards the house and transmitted it live on Facebook.
Police tried to stop the recording, but Ms Ashraf told the BBC that those accompanying her argued that she was doing nothing illegal.
"In Tamil Nadu there is a culture of bullying. There is no space for dissent. If you protest on the streets you are arrested, and if you protest on social media you are charged with defamation. This protest was my way of reclaiming both those spaces," she said.
"Party workers are asking us to give Sasikala a chance, but let her stand for election first."
Ms Natarajan's appointment caused a lot of chatter on Indian social media, both in Tamil Nadu and outside, with many expressing opposition to the decision.
A young Indian rapper has taken on what she calls the "undemocratic" appointment of the new chief minister of southern Tamil Nadu state.