I have read somewhere brown eyes have more melanin than blue eyes and as melanin protects our body from cancers, I thought that brown eyes are more healthier.
People conflate "Dominant" traits to mean "healthier" which is a huge misconception. The previous answer provided a good example of how what is adaptive in one environment can be Mal-adaptive in another. Many Disorders can be dominant for example Huntingtons disease or achondriplastic dwarfism.
Not an answer (I hope an expert provides one), but too long for a comment.
Although I'm unfamiliar with this claim, it's conceivable the optimum melanin content in eyes is subject to a trade-off as it is in skin. In bright areas, skin is dark to prevent damage from ultraviolet light; in dim areas, skin is light to produce enough vitamin D to avoid rickets. Eyes are different in their details, but they too may have adaptive geographical variations (or the distribution of light irises may be due to something more mundane, such as genetic bottlenecks). In theory their optimum colour would trade visual acuity against retinal protection, though I'm dubious about that as light enters through the pupil, not the iris.