Tomatoes are from the Americas, so it wasn't around in European ancient times. I guess it could have been invented in the Americas, but cheese wasn't really a thing there.
Well if you remember Marco Polo, the Venetian explorer from the 13th century, famously established a trade route to China. That was about 250 years before Columbus. Of course there was already trade back and forth between Europe and Asia, but let's just not forget the future Italian who is just famous for connecting with China!
The modern version of a pizza is sometimes said to have been invented in the late 1800s, since that is when the first "proper" recipe is from. But there are a few key points that might inform your question and other answers:
Flat bread baked with toppings already on it has been a thing for several thousand years across the world.
Among which was a type of street food referred to as pizza in Italy before the Tomato arrived in europe
Tomato is from the Americas, it wasn't known and didn't exist in europe until the mid 1500s.
On its own a fresh tomato is not very nurishing compared with other common fruits and vegetables, and since toxic versions exisited, it was mostly used as table decoration in places like Italy for a while.
The first mention of tomatoes as an ingredient in an Italian cookbook is from 1692. And it wasn't widely adopted as a staple food until the 1700s.
There are descriptions from the mid 1800s talking about the wide variety of pizza toppings, as it was already an established food by that point.
Lactose tolerance is not very widespread globally and there's less of it the further you get from north-western Europe, so adding cheese to anything is a surprisingly recent and surprisingly niche culinary choice.
True, which is why dairy farming actually predates widespread lactose tolerance! Nevertheless, you're more likely to get a widespread culture of dairy-eating if you also have a lot of adults who can tolerate a variety of dairy products.
Well, the tomatoes were in the new world, and all the domesticated cows, sheep and goats were on the other side of the atlantic, so that's a minor problem.
I guess you could use like a llama cheese? That might work. Then I'm sure the indigenous people of the Peruvian region had a flatbread of some sort, just about everyone does. Then they just need to harvest up some tomatoes.
So, yeah, they could've. You need to be looking in the Andes in South America though.
The term pizza was first recorded in the 10th century in a Latin manuscript from the Southern Italian town of Gaeta in Lazio, on the border with Campania. Raffaele Esposito is often considered to be the father of modern pizza.
We also have a written account of Persian soldiers using their shields to bake a flatbread with cheese and dates on top from the 6th century BC, which one could count as Pizza.
The baker boy had come to wake up Octavius Picus from his afternoon nap. The delegation from the market at Nuceria had already arrived. They were early. Three people, the boy said, an elderly man with bad eyesight and a young woman guiding him, and another man who was carrying some scrolls. They could be heared chatting with Octavius' wife Marcella in the atrium. No hurry. He sat up on the bedside and rubbed his eyes.
The landlords around Nuceria had recently founded a cooperative and had now come to bargain a new deal for their grain. He wouldn't really know what to make of it, especially since his plan of opening a taverna would certainly require him to buy more wheat and he also wanted to have their beer. Would all that make it more expensive or less? --
Alright, let's serve them something, he told the boy. Make us five of the speciality quick plates, with sausage, eggs and sweet wine, but not the most expensive one. And do not forget the basilicum and cheese sauce on the panem picum!