In popular science fiction, we see habitable Earth-like planets with visible, Saturn-like rings.
In reality, such rings exist only on gas giants, a category where we can find Saturn.
Is it really possible for a planet similar in gravity to Earth to have the kind of gravity needed to create these rings? If no, then what kind of changes would I do to make sure Earth does have them?
We kind of already do:
This is a false-scale image of all objects in orbit around the Earth that NASA/RSA/EASA jointly monitor. You can clearly see the outer ring of GEO objects. Most of these objects are manmade, and most of those are space junk; decommissioned satellites and their debris, pieces of launch vehicles we had to accelerate fast enough to get into a stable orbit, etc. There are thousands more manmade pieces of debris that NASA doesn't keep an eye on because they're too small to pose a re-entry hazard and aren't in a conflicting orbit with any valuable satellites. There was, at one point, an image from a survey probe that caught a beautiful "goodbye" picture of Earth with the sun perfectly aligned to glint off the debris ring around Earth, but I can't find it in the midst of all the "artist's impressions" and enhanced images.
In addition to debris, there's space dust and meteoroids that have been drawn into a stable orbit around Earth, too small to track, but definitely still a hazard; this is what one of those micrometeorites did to the Challenger on STS-7:
As far as why we don't have more, it's mainly chance. A long-term stable orbit around a planet of Earth's size is a fairly tricky thing to achieve, especially without being able to make adjustments. The rings around the gas giants are much more prominent for three reasons:
In addition, what you're seeing around the gas giants is survivor bias; the gas giants, being, well, gas and giants, are much more tolerant of outright impacts, so we don't really see their effects unless we happen to be watching (like when the remains of Shumacher-Levy 9 hit Jupiter). The earth, in its first few million years of existence as a ball of cooling magma, was nearly destroyed by a mass that would become the moon. This chance encounter turned out to be advantageous for the planet, as it gained a sizable shield which has protected the planet from countless impacts over the years by taking the hits itself.
Yes, Earth is large enough to have rings. What you need would be an object orbiting close to the planet like a small moon. It needs to be inside the Roche limit, where the object will start to disintegrate because of the tidal forces.
Earth DID have a ring! BUT, it slowly collapsed in on itself by the force of its own gravity.
4.527–4.533 Billion years ago Theia, a planet with a size similar to Mars, smashed into Earth, and the remains collected in orbit around earth. That is why Luna is comprised partially of mater originating from Earth's crust. (Some people used to call Theia "Orpheus")
Theia originally orbited around Earth at the L4-L5 points. As it gained mass it was slowly knocked loose from orbit by Venus, and caused it to strike Earth, possibly merging the two planetary cores, which is why Earth's core is much larger than the cores of similarly sized planets.
In around a century (less than two) gravity collected all the larger portions together into Luna. Within the next Million years, the crust had formed. Over the next 300 million years (3.9B–4.1B) meteor storms and comets ripped the surface back apart, temporarily creating ANOTHER small ring. It settled back when the "weather" calmed down again, until about 108 million years ago when the last great asteroid impact formed the Tycho crater.
Solid bodies disintegrate after reaching the Roche Limit. However, Earth´s Roche Limit is a little bit inside the atmosphere so any body that is destroyed by tidal forces inside that limit will also fall by loss of potential gravitational energy because of any interaction between air particles and molecules, this is why Earth doesn't have rings. If we have a smaller, less massive and thinner atmosphere it could be possible for Earth to have rings.
