In 1982, the United Kingdom and Argentina fought a major war over the Falkland Islands. And yes, it was a major war. The British lost two destroyers, a landing ship, a merchant vessel, and two frigates. The Argentinean Navy lost a cruiser.
This doesn't count the land or air battles as well. But Argentina hasn't given up hope of taking back what they call the Malvinas. What would happen if they tried to take the islands today?
Both the Royal Navy and the Argentinean Navy have declined greatly, according to the "16th Edition of Combat Fleets of the World." In 1982, the British had two carriers they sent down to the south Atlantic. Today, the UK only has one — the HMS Ocean — and that ship cannot really operate the F-35 fighters that the Brits have bought, and their Harriers are long retired.
The only saving grace is that Argentina lacks the aircraft carrier they had in 1982 (formerly a British carrier, ironically), and has seen its amphibious force cut down to a modified Type 42 destroyer and an old amphibious transport. And its navy has four destroyers, six frigates, and some corvettes, along with three diesel-electric subs.
On the Falklands, the British have a garrison that houses 1,200 soldiers, according to a 2015 London Telegraph report. There is a flight of four Eurofighter Typhoons, plus a tanker and two Chinook helicopters, at RAF Mount Pleasant. A Type 23 frigate is usually in the South Atlantic as well (sometimes, the Brits will send a Type 45 destroyer), and there may or may not be a nuclear-powered submarine there.
The British have contingency plans to reinforce the Falklands, by adding up to a battalion and to send a couple more surface combatants.
This all depends on getting enough advanced warning. Argentina has one advantage: If it can achieve strategic and tactical surprise, its Air Force could use its A-4AR Fightinghawks (A-4s with F-16 avionics) to try and catch the Typhoons on the ground. Similarly, the Argentinean Navy will use its force of Super Etendards to try and sink any British ship in the area.
Once that's done and air and sea superiority has been achieved, the Argentineans would likely move to land troops. The goal would be to grab control as quickly as possible. Once done, Argentina would try to reinforce its garrison before the British can get a sub on scene.
At that point, the British will face the same challenge they had in 1982 – re-capturing the Falklands at the end of a long logistical chain. Subs would move to cut off the Falklands, likely sinking any Argentinean warship or merchant vessel they catch at sea.
There might even be some Tomahawk strikes on Argentinean bases – an effort to catch their planes on the ground. Argentina has 22 Fightinghawks and 10 Super Etendards, according to World Air Forces 2017. If the force can be whittled down enough, the Brits may not need much air cover to take the islands back.
The British would have to commit most of its force of Type 45 destroyers and Type 23 frigates to the attack, and they would need to be able to land troops. Oh, yeah, and do it without carriers or planes. You only have to look at what happened to HMS Prince of Wales and HMS Repulse to see how badly that can go.
But when cut off, and knowing that supplies cannot get through, any Argentinean garrison would eventually agree to surrender, likely parlaying to allow the British to land new troops at RAF Mount Pleasant. Much less hard fighting than in 1982, but the same ultimate outcome – Argentina loses – is the most likely one.