The election will take place at the SDLP's annual conference in November.
Mr Eastwood said his party has had poor election results, was lacking in ideas and he wanted to take it in a fresh direction.
The 32-year-old, who is the SDLP's youngest MLA, said an Irish nationalist party cannot be led from Westminster.
He said Dr McDonnell has "given his life to the SDLP " and would continue to be South Belfast MP.
Mr Eastwood has been nominated by nine party branches to challenge the SDLP leader.
He said his challenge "was not personal " and said politics needed "a new generation".
Mr Eastwood is a former Mayor of Derry and in 2012, he faced controversy after carrying the coffin of a friend who was a former member of the Official IRA and the INLA.
He told the BBC: "Seamus Coyle was a friend of mine and I carried the coffin out of respect."
He said Mr Coyle was on his "own peace process journey".
Mr Eastwood, who has been in the party since he was 14, said the SDLP should "stand for a positive brand of progressive nationalism".
Dr McDonnell has led the SDLP since November 2011
Last week, he told the BBC he did not expect a leadership challenge.
"I don't expect to face a challenge, but if there's a challenge, I'll face it," he said.
Dr McDonnell has faced calls to stand down as leader from a number of party grandees including Brid Rodgers and Seamus Mallon.
The Foyle MP, Mark Durkan, used an interview on the BBC The View programme to call on the South Belfast MP to step aside as leader, arguing that the party leader needs to be at Stormont.

Home Retail Group said that like-for-like sales at Argos fell 1.1% in the eight weeks to 27 February, compared to a fall of 2.2% in the previous quarter.
The electrical and homeware retailer is being targeted by both Sainsbury's and South African retailer Steinhoff.
Both companies have until 18 March to make firm bids for Argos.
Last month, a £1.3bn bid from Sainsbury's for Home Retail was trumped by Steinhoff's £1.4bn offer.
Home Retail recently sold its DIY chain Homebase to Australia's Wesfarmers last month for £340m, leaving Argos as its only remaining business.
Total sales at Argos rose 1.9% to £515m in the eight-week period as demand for furniture and sports goods offset lower sales of electrical goods, such as video games and tablet computers.
The company said Argos opened 90 new stores over the year, bringing the total to 845. The new openings held back the like-for-like sales total over the period, it added.
Chief executive John Walden, who said it had been "another rather eventful period" for the company, was pleased with the improvement in Argos' sales.
Shares in Home Retail closed down almost 0.5% at 179.1p, but have risen more than 80% this year following Sainsbury's interest in Argos.
The supermarket has argued that combining both chains would create a "world-leading" retailer bigger than rivals John Lewis and Amazon UK.
Retail analyst Nick Bubb said: "Home Retail claim that underlying like-for-like sales were broadly flat, once the cannibalisation from the infamous concession inserts in Homebase and Sainsbury are taken out. Gross margins were up, however, by 75 basis points and Home Retail's year-end net cash was strong."
Sainsbury's had offered about 167p a share for Home Retail, but will be forced to at least match the 175p-a-share all-cash offer from Steinhoff, which owns UK furniture retailer Harveys.
However, the supermarket may be reluctant to increase its offer to the level needed to win control of Argos.
Analysts have also questioned the wisdom of Sainsbury's acquiring the struggling retailer.
The Social Democratic and Labour Party (SDLP) MLA Colum Eastwood has confirmed that he is set to challenge Alasdair McDonnell's leadership of the party.
[NEXT_CONCEPT]
Argos, the retail chain at the centre of a bidding war, has shown a fall in sales over the Christmas period.