The double world champion has the full approval and support of McLaren and engine partner Honda, who are having a difficult season in Formula 1.
Alonso, 35, will race for the Honda-powered Andretti team on 28 May, and the car will be branded a McLaren.
McLaren are yet to decide who will replace him in Monaco that weekend, but Jenson Button is a possibility.
The 2009 world champion has retired from F1 but is contracted to McLaren as an ambassador. It is not known whether the Briton would want to come back to drive an uncompetitive car.
Alonso said he had long held an ambition to win the so-called 'triple crown' of Monaco, the Indy 500 and Le Mans.
Only one man has won all three in his career - the late Graham Hill in the 1960s.
Alonso, who won the Monaco Grand Prix in 2006 and 2007, said: "It's a tough challenge, but I'm up for it.
"I don't know when I'm going to race at Le Mans, but one day I intend to. I'm only 35. I've got plenty of time for that."
The Spaniard added he would definitely race for McLaren for the rest of the season, dismissing speculation he could quit part way through the year because of the Honda F1 engine's poor performance.
"It's of course a regret that I won't be able to race at Monaco this year," he said. "But Monaco will be the only 2017 grand prix I'll be missing, and I'll be back in the cockpit for the Canadian Grand Prix in Montreal in early June.
"I've never raced an IndyCar car before, and neither have I ever driven on a super-speedway, but I'm confident I'll get to grips with it fast.
"I've watched a lot of IndyCar action on TV and online, and it's clear that great precision is required to race in close proximity with other cars on the far side of 220mph [354km/h]."
Alonso acknowledged he would be on "a steep learning curve".
But he added: "I'll be flying to Indianapolis from Barcelona immediately after the Spanish Grand Prix, practising our McLaren-Honda-Andretti car at Indy from 15 May onwards, hopefully clocking up a large number of miles every day, and I know how good the Andretti Autosport guys are.
"I'll be proud to race with them, and I intend to mine their knowledge and expertise for as much information as I possibly can."
McLaren have supported Alonso's wishes because they recognise the efforts he has been putting in - and the frustration he is feeling - after three uncompetitive seasons since joining the team in 2015.
McLaren executive director Zak Brown said: "Could Fernando win this year's Indy 500? Well, I wouldn't be so silly as to make any such rash prediction, but I expect him to be in the mix.
"Put it this way: the team he'll be racing for won the race last year, using the same Honda engine, and he's the best racing driver in the world. That's quite a compelling combination.
"He'll have his work cut out to acclimatise to running at super-speedway velocities, but ultimately it's quality that counts in all forms of motorsport, and Fernando is very definitely quality. He's ballsy and brave too."
Alonso joined McLaren-Honda with the intention of winning a third world title, but the package has been uncompetitive, with the vast majority of the blame lying with the Honda engine. His best results have been three fifth places.
He has won 32 grands prix - sixth in the all-time list - but has not stood on top of the podium since the 2013 Spanish Grand Prix in a Ferrari.
Alonso's contract runs out at the end of this season. McLaren want him to re-sign and there is a hope this will help persuade him to do so.
The unexpected development marks McLaren's return to the Indy 500 for the first time in 38 years. They won the race with their own car in 1974 and 1976.
IndyCars is now a 'spec' formula, where all teams use the same car, though the different engine manufacturers are allowed to design their own aerodynamic bodykits.
The Indy 500 is the most prestigious race in the USA and the blue riband event of the IndyCar Series.
The unique practice schedule gives Alonso more than the usual amount of time to prepare for a race.
There is a full week of practice, six hours a day for five days, before the qualifying weekend on 20-21 May, and two more days of practice before the race on 28 May.
The Andretti team, run by former IndyCar and grand prix driver Michael Andretti, is one of the leading teams in the championship and won the event last year with American Alexander Rossi, who raced five times in F1 for the Manor team in 2015.
Andretti raced for McLaren in F1 in 1993 as Ayrton Senna's team-mate, completing 13 races with one podium finish before being replaced by Finn Mika Hakkinen and returning to race in the US.
Andretti said: "Fernando's lack of experience on super-speedways is not of concern to me.
"I do believe that the Indianapolis 500 is one of the best places for a rookie to start because there is the opportunity for so much practice time on the track - and, as we have demonstrated, it can be won by a rookie.
"Fernando is a great talent and I have full confidence that he will represent very strongly for McLaren, Honda and Andretti Autosport."
This is a bold and exciting move by Alonso but one with plenty of risks.
The Spaniard remains in the very highest echelon of Formula 1 drivers and has more than enough talent to succeed in any car, but the type of racing he will encounter at Indianapolis is unlike anything he has yet experienced.
If he was racing on a road course - what Americans call F1-type tracks - he would be expected to be absolutely competitive straight away.
But Indianapolis is a so-called 'super-speedway' - an ultra-fast oval track where average lap speeds can exceed 230mph.
Not only does Alonso have to get used to the intricacies of racing on a banked oval, including all the technical challenges involved, he will also have to learn the art of the 'draft' - using the slipstream of another car to gain speed - which is critical to oval racing.
And because of the high speeds involved and the proximity of the cars, IndyCar racing has a reputation for being notably more dangerous than F1, and any accident can have serious consequences.
But what he is doing is not without precedent.
Hill, Jim Clark and Emerson Fittipaldi all won the F1 world title and the Indy 500, while Nigel Mansell switched to IndyCars in 1993 after failing to agree terms with Williams following his title success in 1992.
Alonso will be able to count on advice not only from team owner Andretti, but also team-mate Takuma Sato, who raced in F1 in the mid-2000s.
And former Indy 500 winner Juan Pablo Montoya, who is also in this year's field, also raced against Alonso in F1 from 2001-2005.
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McLaren driver Fernando Alonso will miss the Monaco Grand Prix in May so he can race in the Indianapolis 500.