In his first televised speech, an emotionally-charged Abbasi dismissed the corruption allegations against Sharif as baseless and said he hoped the deposed premier would return to parliament soon.
He said the people of Pakistan did not accept Sharif's disqualification, and vowed to follow in his footsteps of Sharif.
Abbasi stopped short of criticizing the judiciary, saying legal experts were surprised by the decision to disqualify Sharif and that "no one is willing to accept it." He said Sharif was victimized for putting Pakistan back on the path of progress and bringing in foreign investments worth billions of dollars to Pakistan.
He said Sharif was able to attract foreign investment because he was an honest person.
Abbasi also asked his countrymen to pay their taxes honestly as he plans to take stern action against anyone involved in tax evasion. He said rich people must pay their taxes for Pakistan's economy to improve. He promised better health and education facilities for his countrymen.
Abbasi said he would ensure the rule of law and that he would run the country strictly according to the constitution.
Sharif's party has said it will file a petition with the Supreme Court next week asking for a review of its decision to disqualify him.
Sharif's party wants Abbasi to serve as an interim premier until Sharif's younger brother, Shahbaz Sharif, the chief minister of Punjab province, wins a national assembly seat in a by-election.
The opposition has criticized this intention as dynastic and undemocratic. The next general election is due to take place in June 2018.
Abbasi's rival candidates Qamar and Ahmed congratulated him on becoming the new prime minister.
Meanwhile, a female lawmaker from Khan's party on Tuesday quit the party, calling the party leader a "characterless person."
Ayesha Gulalai Wazir posted a tweet saying the honor of female party members was not safe because Khan had an "immoral character."
There was no immediate comment from Khan's party about the allegation.