It notes that Mr Varadkar joins Luxembourg's Prime Minister Xavier Bettel as the only other openly gay world leader now serving.
He added that his father - an Indian immigrant - would be proud that his son could be "judged by his actions, not his origins or identity".
After the election, the center-right politician promised the Dail, "I've been elected to lead but I promise to serve". "In the months ahead, I will seek to do exactly that".
Mr Kenny's speech marked his long-awaited resignation as leader of the Republic's minority government after six years in the role.
Varadkar went up against housing minister Coveney in the leadership race, who bagged two-thirds of the party membership votes but lost out to the more popular candidate in the end.
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This clearly shows that it was no random comment; it was a pre-planned political move by the Congress Party. Sonia Gandhi in particularly, come out and disown (it). and apologize", she added.
His win was hailed by LGBT campaigners.
Mr Kenny is also the only Fine Gael leader to be re-elected taoiseach.
European Council President Donald Tusk, tweeting words of congratulations in the Irish language, assured Varadkar that his fellow EU leaders shared a "great sensitivity" about the challenge the island of Ireland faces in the Brexit talks.
"There has been a 100% increase in the number of young people accessing our services since the referendum".
More as we have it. "The role of Taoiseach is profoundly different from that of being a departmental minister and party spokesman".
Mr Varadkar also warned his colleagues not to book summer holidays yet, as he may request the Dail to sit a week later than expected.