In a perfect world, you would have provided us with more information about a specific public school. For example, it would be helpful to know how highly a public school is ranked in the state, the number of students per class, the relative strength of each academic department, how students perform on state tests, how they perform on tests like the SAT, whether the school offers APs...how many? And so on. There are many variables and features that define one public school from another. Some are excellent at college preparatory work and can go toe-to-toe with the nations most elite private schools. Others are rather mediocre and it's up to the student to make the most of his or her studies and experience.
In evaluating the question above, you need to ask yourself, what kind of public school is being assessed, and how rigorous is the curriculum? This will give you insights into the readiness for college that a particular public school can provide.
Some years ago a friend of mine chose to have her son leave public school after his high school sophomore year, and complete his last two years at a private school. Her thinking was simple. Her son wanted to study science at an elite college. The current public high school had a weak science department. Also, although this public school was ranked number #1 in the state at the time, classes were maxed out at fire code. A typical AP class had 30 kids in it. She wanted her son in a smaller, more nurturing environment where he would truly thrive.
The private school they chose accepted this new student, but made a fuss and wanted the student to repeat his sophomore year. The private school felt the public school had done a poor job of preparing this student for the academic rigor at the new school. They were concerned he would trail behind his peers. I remember having a private chuckle at this with the boys mother. The public school he was coming from was killer; it was more than on par with the private one. This student was ready to tackle tough academics. He was used to an incredibly demanding workload and high standards. He had grit. He had skills. In fact, the literacy department at his public school was standout; he already knew how to compose a cohesive and persuasive essay and write a thesis.
The parents refused to have their son repeat sophomore year. So the student was allowed to attend the new private school as a junior. Within months he had won several academic awards, become a star pupil and earned the respect of the faculty. More importantly, the private school sheepishly acknowledged the prior school had done an exceptional job of preparing this student for academic success.
The point of is, a public school can well prepare a student for college and challenging academics. And so can a private one. How well that student is prepared will depend on the actual school and It's resources, teaching staff and mission. Of course, it will also depend on the student.
Public schools have some advantages over private ones in that they are held to state standards. You should be able to access that information. Also, Ivy League schools tend to admit more students from public rather than private schools, though this may have less to do with the caliber of public school students and more a desire to not over advantage already privileged children.
At a minimum, public schools should provide college bound students with the opportunity to develop strong study skills, time management skills, competencies in math, science, literacy and a foreign language (most colleges require three years). There should be an opportunity to take AP classes. Students who take an AP class and then go on to score a 4 or higher typically receive college credit (though not always. Some universities are persnickety about what they will accept). Public school should also be a time when a student learns how to manage more responsibility, perhaps serve in leadership positions, join clubs that reflect a growing range of interests, and participate in serious athletics where teamwork is stressed. These are all experiences that foster readiness for the future.
Again, how well this is done, depends on many circumstances, including ones future goals. It's important to consider the type of college one hopes to attend. Being ready to succeed at a top flight college where everyone graduated at the head of the class, may be different than being ready to perform well at a local state university.
I hope this answer provided you some guideposts.