Republican Jesse Kelly, who lost Tuesday’s special election to replace former Rep. Gabrielle Giffords (D-Ariz.), will not run again in the regular election against Rep.-elect Ron BarberRonald (Ron) Sylvester BarberPrinciples and actions mean more than Jeff Flake’s words Giffords to lawmakers avoiding town halls: 'Have some courage' Ten House seats Dems hope Trump will tilt MORE (D-Ariz.).

"Looking at the results from Tuesday, we have decided to withdraw from the race for Congress in AZ-02 and to seek other opportunities," Kelly said in a statement. "I will forever be thankful to our generous supporters and volunteers."

Kelly’s decision, first reported by The Hill, clears the way for retired Air Force Col. Martha McSally to clinch the GOP nomination to take on Barber.

On the Democratic side, Barber will face a primary challenge from state Rep. Matthew Heinz.

Barber, a former Giffords aide, defeated Kelly by almost 7 points on Tuesday and will be sworn into office next week. But Barber can only count on being in Congress for the next six months.

The timing of Giffords’s resignation in January set off a complex sequence of events that one Arizona operative described as a Rubik’s cube. A special primary and general election were held to finish Giffords’s term, which expires at the end of the year. In August — just two months after the special election — voters in southern Arizona will return to the polls for another primary, followed by a general election in November for the full term starting in 2013.

Those contests will be held under new congressional lines that were not in effect for the special election. The old district was competitive for both parties, but leaned slightly Republican. The new map makes the district about 3 points more Democratic, giving Barber an even greater edge.

Adding another dimension to the Rubik’s cube was the filing period for the regular election, which ended before the special election had occurred. That means no additional candidates can file to take on Barber.

McSally, who came in second out of four Republicans in the special primary, has filed for the fall race, as has Republican Mark Koskiniemi. But Koskiniemi is a political unknown, and Republicans in Arizona said they expected McSally to coast to the nomination. McSally’s campaign did not respond to a request for comment.

Barber heads into the regular election with all the advantages of incumbency, plus the added distinction of being the newest member of Congress. Democrats have spent the few days since Barber’s victory lauding him for holding onto the seat Giffords vacated in January, one year after a gunman shot both Giffords and Barber and killed six others.

But Democratic luminary or not, Barber won’t have the primary to himself.

Early in 2012, a number of other Democrats had declared their intentions to run for Giffords’s seat, but cleared the field after Giffords asked Barber to run to replace her and endorsed his campaign. At the time, Barber wouldn’t say whether he would also seek a full term, leading some Democrats to believe he would serve as a placeholder for Giffords, then bow out of the regular election.

But Barber decided to seek a full term, and one of those Democrats — Heinz — decided there was no reason to cede the regular election to Barber.

"I'm really pleased and congratulate Ron for his win," Heinz told The Hill on Thursday. "I think southern Arizona made the right choice to complete Gabby's term. Now we have to have a discussion with the voters of southern Arizona as to who the best advocate will be for the long term."

With the Democratic establishment lined up firmly behind Barber, support in the primary will be difficult for Heinz to come by. Democrats in Arizona will almost certainly pressure Heinz to drop his bid, averting a divisive primary that could weaken Barber’s prospects for holding onto the seat.

“There’s a sense that his fundraising — whatever fundraising there was — is going to hit some brakes,” said one Arizona Democratic operative.

But Heinz’s campaign manager, Evan Hutchison, pointed out that Heinz was the first candidate to declare for the race, and said Heinz would consolidate support and donations from other donor bases. Heinz, an openly gay physician at a Tucson hospital, will court support from the medical community and the gay and lesbian community, Hutchison said.

Democrats say they are encouraged by Barber’s victory on Tuesday, in which he defeated Kelly by about 13,000 votes. In 2010, when Kelly challenged Giffords amid a major GOP wave, Giffords won by about 4,000 votes, meaning Barber did even better against Kelly than his former boss did two years ago.

The Democratic Congressional Campaign Committee has argued that the party’s success in the special election portends further victories in November in two other Arizona districts: a newly created toss-up seat in Phoenix and an open seat in northeastern Arizona where former Rep. Ann KirkpatrickAnn KirkpatrickGold Star father attacked by Trump steps up role in Dem primaries House Dems highlight promising new candidates Vulnerable House incumbents build up war chests MORE (D) is running to reclaim the seat she lost in 2010 to Rep. Paul GosarPaul Anthony GosarHouse rejects effort to condemn lawmaker for demanding 'Dreamer' arrests Hispanic Dems seek vote to condemn GOP lawmaker for demanding arrests of 'Dreamers' High-ranking Trump official attends hunting convention MORE (R).

A pickup of either of those two seats would put Democrats closer to the 25 House seats they need to flip in November to reclaim control of the House.

But Republicans claim the special election was just that — special — and that the unique circumstances of the highly emotional process to replace Giffords make it impossible to extract broader conclusions about the state of the race in Arizona or elsewhere.

- This post was updated at 4:15 p.m.