Can Pakistan destroy an Indian aircraft carrier during a war?

Answer 1:-

No, Pakistan cannot destroy an Indian Aircraft carrier.
INS Vikrant (IAC-I) is the first aircraft carrier built in India and the first Vikrant-class aircraft carrier built by Cochin Shipyard (CSL) for the Indian Navy.
Sensors and Avionics
INS Vikrant has integrated with Selex RAN-40L AESA radar and EL/M-2248 MF-STAR AESA(Multi-Function Surveillance, Track And Guidance Radar)  which is used for long range maritime air surveillance and early warning. The radar uses a fully solid state active phased array antenna and capable of tracking and detecting air targets like aircraft or drones up to 400 km away.
It is also equipped with C/D Early air-warning radar used primarily for the long-range detection of its targets, i.e., allowing defenses to be alerted as early as possible before the intruder reaches its target, giving the air defenses the maximum time in which to operate

(Otobreda 76 mm (3in) naval artillery gun)
The INS Vikrant is being armed by latest arms which includes 4x Otobreda 76 mm (3in) naval artillery gun which have a high rate of fire and availability of specialised ammunition make it well-suited to varied roles such as short-range anti-missile point defence, anti-aircraft, anti-surface, and ground support. Specialised ammunition includes armour piercing, incendiary, directed fragmentation effects, and a guided round marketed as capable of destroying manoeuvring anti-ship missiles
Vikrant is also armed with Barak 1 & Barak 8 Surface-to-air missile launchers (2×32 cells VLS) which are specially used to defend against any type of airborne threat including aircraft,helicopters,anti-ship missiles ,UAVs,cruise missile and combat jets.
(Barak 8 Missile armed in INS Virant)
The AK-630 fully automatic naval close-in weapon system based on a six-barreled 30 mm rotary cannon also defend the Vikrant against anti-ship missiles and other precision guided weapons.
(AK-630 fully automatic naval close-in weapon system)
Aircraft Carried on the Deck
The deck is designed to enable aircraft such as the MiG-29K to operate from the carrier. It is expected to carry an air group of up to thirty aircraft, which will include up to 30 fixed-wing aircraft  primarily the Mikoyan MiG-29K and the naval variant of theHAL Tejas Mark 2, besides carrying 10 Kamov Ka-31 or Westland Sea King helicopters. The Ka-31 will fulfill the airborne early warning (AEW) role and the Sea King will provide anti-submarine warfare (ASW) also carried HAL Dhruv on the deck.
Sea control against Pakistan
In establishing sea control across the northern Arabian Sea, the Indian Navy would fight a tricky battle in coastal waters against the Pakistan Navy. The latter, outnumbered and outgunned, knows it would get quickly wiped out on the open seas. It is likely, therefore, to withdraw close to the Pakistan coast where the Pakistan Air Force (PAF) would provide it air cover.
To close in with this fleet, India's CBGs must have the air defence capability to beat off the PAF. Key to this would be the MiG-29K fighter, flying from aircraft carriers; and air defence systems like the Barak, and the much-awaited new Long Range Surface to Air Missile. The LR-SAM, which the Defence R&D Organisation (DRDO) is developing with Israel, will be deployed on warships by end-2015. These missiles would also protect the CBG against anti-ship missiles - like the Harpoon and Exocet - fired from Pakistani submarines, warships and aircraft.
"The Israeli Barak missile, which was bought in 2001, for the first time provided the Indian Navy with genuine air defence capability. The LR-SAM will make air defence even more reliable," asserts Chauhan.
Until the LR-SAM is operational, Indian warships remain critically vulnerable to air and missile attack, but the navy believes it will be worth the wait. "This (delay) is the price that you pay when you go in for high-tech, state-of-the-art systems", says Vice Admiral Satish Soni, who heads Eastern Naval Command.
The LR-SAM will also defend Indian warships against a feared ocean predator - long-range maritime patrol (LRMP) aircraft like Pakistan's P3C Orion, which will fly 12-hour missions from Karachi to scour the seas, locate Indian warships, and launch anti-ship missiles from 50 kilometres away.
The LR-SAM's 70-kilometre range lets it engage the LRMP aircraft even before it launches its anti-ship missile. If the aircraft manages to launch a missile, the LR-SAM is designed to shoot it down before it strikes a warship. For the LRMP aircraft, an attack on a CBG would be suicidal. Its presence betrayed by the launch of a missile, MiG-29Ks fighters scrambled from an aircraft carrier would quickly overtake it and shoot it down.
After coming within range of Pakistan's surface fleet, Indian warships would launch an air-sea attack - striking Pakistani warships with anti-ship missiles like the Brahmos, from ranges of up to 300 kilometres; and with fighter aircraft launched from the aircraft carrier.
Detracting from India's convincing naval superiority in the Indian Ocean region is only its vulnerability to enemy submarines. This stems not just from a depleted submarines force, but also neglect of the capability to detect and destroy enemy submarines.
Answer :-2
Short Answer : No
Long Answer : Aircraft Carriers do not travel alone, they are always escorted/protected by other ships.
First of all it will be very difficult for Pakistani Navy to find where the carrier actually is, not many people know the exact position of an aircraft carrier for all we know it's somewhere in 73.56 million km²area of the Indian Ocean.
Secondly, aircraft carriers have destroyers, frigates and submarines protecting it in addition to multiple anti-missile, anti-aircraft and anti-ship systems already installed on the carrier, not to forget a squadron of fighter aircraft's on-board.
During war security would be increased multiple times, a no fly zone will be enforced several miles around the carrier, several hundred miles of airspace around the carrier will be heavily monitored.
Let's assume a war starts with Pakistan tomorrow and INS Vikramaditya is deployed to enforce a blockade of Karachi.
Any attack by Pakistani air-force will first have to deal with this baby.
This Beauty is a Kolkata class stealth destroyer, it is equipped with state of art IAI EL/M-2248 MF-STAR AESA multi-function radar, Thales LW-08 D-bad air search radar, IAI EL/M-2238 L band STAR surveillance radar (don't worry i also don't know what that means), supersonic BrahMos missiles (BrahMos can hit a target 300km away in less than 5mins, that's all you need to know), INS Vikramaditya in wartime will most likely be escorted by two of these beauties.
In addition to 2 destroyers the aircraft carrier will most likely be escorted by at least two frigates, few smaller vessels and a few submarines.
Oh did I forgot to mention these beauties?
It's a MIG-29K, a 4.5 Generation multi-role fighter aircraft which is more than capable of intercepting any aircraft in Pakistani air force arsenal, any strike by Pakistani air force is likely to encounter at least 20 of these bad boys (there are 30 on the aircraft carrier) before they are even in the 200 miles radius of the carrier.
Let's suppose that Pakistani air force 'talented' pilots and their 'ultra modern' aircraft are able to get past these defenses, then hundreds of these anti-aircraft missiles will be chasing them.
So yeah, there's absolutely no chance of Pakistan ever sinking any Indian Aircraft Carrier.
Pakistan has already tried sinking INS Vikrant but ended up losing their own leased submarine.
By the way, nuclear weapons are useless against ships; they could never hit a ship and by the time they strike nearby waters the ships will have enough time to get into a defensive position, contrary to popular belief nuclear weapons do not result in an apocalyptic winter.
People think INS Vikramaditya travels like this,
But let's zoom out a bit.
There you have it.
It is not as grand as that of the United States,
but we will get there....eventually :P