No, Deathtouch will not destroy an Indestructible Creature. The reason is the way the two abilities work.

Deathtouch

According to Comprehensive Rules 702.2, Deathtouch is a Static Ability, which means it's always on as long as the Creature has it.

Rule 702.2b states:

"A Creature with toughness greater than 0 that's been dealt damage by a source with Deathtouch since the last time state-based actions were checked is destroyed as a state-based action."

It's a little awkwardly worded, but that means if a Creature that's alive has been hit by a Creature with Deathtouch (whether Attacking or Blocking), it will be Destroyed and put in the Graveyard.

The rules go on to explain that any nonzero amount of Combat Damage by a Creature with Deathtouch is enough to be considered lethal damage. That means if it does at least one damage, then it's lethal.

Indestructible

Rule 702.12 covers Indestructible, which is also a Static Ability. Rule 702.12b says

"A Permanent with Indestructible can't be destroyed. Such Permanents aren't destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage."

In other words, you can't destroy an Indestructible Creature.

How Deathtouch Interacts With Indestructible

When a Creature with Deathtouch meets a Creature with Indestructible in Combat (or, frankly, if you figure out another way to have a Deathtouch Creature deal damage to an Indestructible Creature), the Creature with Deathtouch will deal damage to the Indestructible Creature.

That damage will be marked on the Indestructible Creature, then when the State-Based Action occurs to see if the Indestructible Creature has survived, Indestructible will prevent the lethal effect of the Deathtouch.

Corner Cases

A few other effects can change the way these two abilities interact. Let's go through a few of them.

First Strike

If a Creature with Deathtouch has First Strike (or Double Strike, which means it deals both First Strike and regular Combat Damage), it will deal its damage before a Creature without First Strike.

This can make for a very strong Attacker or Blocker, because most Creatures will simply die to the Deathtouch damage before having a chance to deal damage of their own. Glissa, the Traitor, which naturally has First Strike and Deathtouch, is a great example of this.

However, if the Creature it's meeting in Combat has Indestructible, the Creature with Indestructible still can't be destroyed, so it doesn't matter. The damage will be marked, drop off, and regular Combat Damage will proceed.

Trample

There are no Creatures currently with both Deathtouch and Trample, but should a Creature gain both, the interaction is rather interesting. Trample states in Rule 702.19b that an Attacking Creature with Trample first assigns damage to the Creatures blocking it, but once all those Blocking Creatures have been dealt lethal damage, any excess damage is assigned to the Blocking Creatures' controller.

Because Deathtouch reduces the amount of damage required to be lethal to just one, the Attacking player can assign one damage to the Blocking Creature and have the rest Trample over. Force of Nature, an 8/8 with Trample, Enchanted with Aspect of Gorgon, an Aura which grants Deathtouch, would be an example of this.

A funny thing happens when the Blocking Creature has Indestructible, though. Trample only cares if the damage is lethal, not if the Creature actually dies.

So, the Attacking Creature with Trample and Deathtouch can deal one damage to the Blocking Creature and Trample the rest over. The Indestructible Creature will take its one damage, shrug it off and stay alive, while the rest of the damage Tramples over to the Blocker's controller.

Getting Rid of Indestructible Creatures

Indestructible Creatures can be removed in several ways. The following are most common ways to get rid of an Indestructible Creature.

Reduce Toughness to Zero

A Creature with Toughness of 0 or less will automatically be put into the Graveyard as a State-based Action as soon as its Toughness becomes 0 or less. A card like Tragic Slip, which gives -1/-1 to a Target Creature but -13/-13 if another Creature has died this turn, is a great example; it doesn't Destroy, it reduces Toughness.

Exile

Exiling a Creature with a card like Path to Exile (which Exiles a Target Creature and allows that Player to search for a Basic Land) will Exile a Creature with Indestructible. Exiling isn't the same as Destroying, and Indestructible only prevents a Creature from being Destroyed, not Exiled.

Sacrifice

Forcing a Player to sacrifice a Creature with Indestructible will put the Indestructible Creature in the Graveyard. A card like Chainer's Edict, which says "Target Player Sacrifices a Creature," can force a Player with only Indestructible Creatures to Sacrifice one.

Final Answer

No, a Creature with Deathtouch will not destroy an Indestructible Creature; Indestructibility will prevent the lethal effect of the Deathtouch.