Short answer. Yes. Article 674 obliges a building owner to construct his roof so that rainwater falls on his own land, or on a street or public place, and not on his neighbour's. He must also collect the water even on his own side so that it causes no damage next door.
What the law says
The owner of a building shall be obliged to construct its roof or covering in such manner that the rain water shall fall on his own land or on a street or public place, and not on the land of his neighbor
Civil Code, Article 674 — Rain Water From Roofs. Read the full provision →
The duty is on the design, not on the damage
Read what the obligation attaches to: the roof must be constructed so the water falls on his own side. That means the breach is complete when the roof discharges onto your land, whether or not you can show the water has harmed anything. You are not required to wait for erosion, a damp wall or a flooded yard before objecting. This matters practically, because the cheapest moment to fix a roof line is while it is being built, and an owner who is told early cannot later say he was never given the chance.
Collecting the water is a second, separate obligation
Redirecting the discharge onto his own ground is not the end of it. The article adds that even where the water falls on his land, the owner must collect it in such a way as not to cause damage to the adjacent land or tenement. So a downspout that empties beside the boundary and lets the flow run straight across is no compliance at all. Concentrating what used to be dispersed runoff and delivering it to one point on your line is the classic version, and it usually does more harm than the original eaves did.
Shared ownership next door is no excuse
The article deals expressly with the case where the adjacent land belongs to two or more persons, one of whom owns the roof. Being a part-owner of the land the water lands on does not entitle him to discharge onto it. The point comes up constantly among co-heirs and in family compounds, where the person who built the structure assumes his share in the property covers it. It does not — the other co-owners are neighbours for this purpose, and their consent, not his own stake, would be what changed the position.
Document the flow, then ask in writing
Rain is the evidence, so take photographs and video during an actual downpour showing where the water leaves the roof and where it lands; a dry-day photograph of a gutter proves very little. Record the wall staining, the erosion channel or the pooling, and note when it started. Then write to the neighbour describing the problem and what you want done, and keep proof of delivery. Most of these are solved by a gutter and a downspout, and a letter that arrives before the damage is serious tends to get one.