Is a fatter tire likely to get fewer flats
The two most common types of flats are
1) pinch
2) puncture 

Can also have the spoke puncture the tube from the inside but nothing a tire can do about that.   

A fatter tire is more resistant to both pinch and puncture flats.

A larger tire will also have softer ride (due to lower pressure) and better traction due to more tire in contact with the road.  This durability and comfort come at a cost.  A larger tire is heavier, more wind resistance, and more rolling resistance.

Wheel sizes are reported in a number of ways.
The actual actual wheel size or actual wheel size plus nominal tire size.  

For sake of fair comparison I will use actual wheel size.
For example ISO 622 is the same as both 700mm and 29".   

Will compare ISO 622 wheel diameter and tire diameter of 1" versus 2".
Same rubber in each tires and same thickness.
A fatter tire will tend to be thicker but for the sake of head to heads comparison assume same thickness

The tire must support the weight of the rider and bike.
If I am 170 lbs and 60% of the weight is on the rear wheel then let's call it it 100 lb on the rear wheel.  

The 2" diameter tire will run at a lower pressure.
Let's assume the 1" tire runs at 100 PSI and the 2" at 60 PSI.  

A bigger tire runs at lower pressure because it has more contact area.
In this example the 1" tire will have a contact area of 1 square inch.
And the 2" tire will have a contact area of 1.67 square inches. 

Pinch Flat

Pinch flat is when the tube is pinched against rim and splits open.
When you hit a blunt object such as a curb.

A larger diameter tire is more resistant pinch flat for two reasons 
 1. With the larger diameter the outer tire (starts) further from the curb.
 2. Due to the bigger diameter the  fatter tire builds surface area faster so it does not need to deform as far to match the force of the curb.

Puncture Flat

A sharp object penetrates tire and tube.

Identical rubber will have the same force per surface area to penetrate.
The rubber does not know if it has 1" or 2" or buddies.      

Assume a triangular piece of glass 10mm by  10mm by 10mm.    

Along comes this this fat soft tire.  The tire deforms and the rubber makes contact with the road around the glass like the glass is not even there.   Because the tire deformed and made contact with the ground the glass never exceeded the minimum force to penetrate the tire. 

Even if the fatter tires comes in contact with more glass is still distributes the load.
Would you rather stand on a bed of nails of lie on a bed of nails.   

Then along comes a hard shinny skinny tire that cannot absorb the glass and at the point it is just tire on glass there is sufficient pressure for the glass to penetrate the tire.  The higher pressure tire did deform but not as much as much as the lower pressure tire.  In this example when the tire was directly on the glass no tire was in contract with ground to take the part of the load.


A smaller piece of glass in which both tires deform and touch ground around the glass the softer (lower pressure) tire is always going to have more tire on ground around the glass and take more weight off the glass.  The large tire is going to take enough weight off the glass to avoid penetration for a range of heights and sharpness.

This is not an artificial condition.  A glass bottle is about 4 mm thick. The small sharp shards of glass from the bottle are 4 mm.

Even a piece of glass that penetrates both tires the glass will penetrate the harder (higher pressure) tire further.  Because of deformation the glass will not be pushed in all the way.  Again the softer tire has more deformation.  So could have a case where the glass penetrates the tube on the hard tire but not the softer.  

If you ride the same piece of glass long enough it will eventually works its way into both tires.  So inspect your tires for glass.  On a regular basis.  For sure every time you air them.

Tire Pressure

Tire pressure effects both pinch and puncture flats (and rolling resistance).

Increase pressure reduces pinch flats.

Increased pressure increases puncture flats because the tires does not deform as much (spread the load).

So there is a sweet spot.  Go with the manufacturer for that sweet spot.  Tire will have a pressure range.  If you weigh 180 of more then use the maximum.  If you wight 140 of less then use the minimum.  In between use liner interpolation.   If you weigh 160 you are 1/2 between and should use the pressure 1/2 way between.

Mountain bikers like to run at lower pressure for more contact (traction) and to absorb  roots and rocks.  If you are running at a lower pressure then be aware and don't bang on roots and rocks as hard.  

Tubeless

Tubeless tires offer the advantage of no tube to pinch so no pinch flats.  Can run them at a lower pressure.  A tubeless tires with sealant is not less susceptible to a puncture flat but it will seal small punctures.