Why do modern ships use diesel engines over petrol engines?

There are many reasons why Diesel is a better choice for marine applications:

Reliability - Diesel engines ignite the fuel by compression rather than spark. There is no HT (high voltage electrical) system so they are much less sensitive to moisture. In fact, a Diesel will usually run perfectly happily partially submerged in water. This is important on a boat.

Thermal efficiency - big two-stroke Diesels are very efficient.

Fuel flexibility - a Diesel engine can run on a wide variety of fuel sources and is not fussy about quality. Large ships often use the cheapest heavy oil they can.

Safety - heavy oils are much more stable than more refined fuels and therefore more difficult to set on fire.

Higher energy density - Diesel fuel holds more energy by volume and weight.

Moreover - 

• Diesel engines have higher thermal efficiency. A very large marine diesel can exceed 50% thermal efficiency. Gasoline engines in practice rarely reach 35%.
• Diesels, especially big diesels, are quite forgiving about fuel quality. The fuel used by ships is often the cheapest, most viscous, and most polluting liquid fuel available, something that can’t be used for any other purpose. Petrol (gasoline) engines require one of the most expensive liquid fuels.
• Because very large petrol engines have rarely (never?) been built, the engineering of vary large diesel engines is much better known. The most powerful marine diesel today is about 109,000 horsepower. The most powerful petrol engine ever built was probably a 36-cylinder Lycoming aero engine which produced about 5,000 horsepower.
• Fuel usually used by diesel engines has a much higher flash point than petrol. Petrol (gasoline) has a flash point below -40C. Diesel fuel may have a flash point over 90C. This makes it a much safer fuel because it is really hard to set on fire accidentally.
• Torque is the champion here. High torque is needed to move heavy loads. If comparing a petrol engine to an equivalent diesel engine, the diesel will always have higher torque. The higher torque comes from the need for a higher compressing ratio needed for compression ignition. To achieve the higher compression ratio a longer stroke is required. The longer stroke comes from a greater crankshaft offset. This offset gives greater torque.
• Another aspect is that diesel engine can make tremendous torque at very low RPM. Very simply put, more fuel equals more torque when everything else is kept the same. In a diesel the amount of fuel added is what controls the power. The throttle controls how much fuel is added. This means that a diesel always runs lean. At idle the engine uses hardly any fuel. This lean mixture allows for the addition of large quantities of fuel even at low RPM.
• The only real draw back to this torque production is a limited RPM. This is compensated by a gear box with lots and lots of gears.

If a gasoline engine was used it would have to be much larger. The much larger engine would make for greater fuel consumption.