In Exodus, the sixth plague of Egypt was a plague of boils (שְׁחִין), the only instance that God brought a disease on the Egyptian people themselves - apart from the inexplicable death of their firstborn, but this is something that would not normally be called a disease. In reading Exodus 15:26, it would appear that God is saying that if the Israelites "diligently hearken to his voice" he would not inflict this plague upon them. However, a parallel verse in Deuteronomy is more explicit about what will be done if the people obey his commandments:
Deuteronomy 7:15: And the LORD will take away from thee all sickness, and will put none of the evil diseases of Egypt, which thou knowest, upon thee; but will lay them upon all them that hate thee. [my emphasis]
The Benson Commentary recognises the ambiguity of Exodus 15:26, saying the diseases in question are "Either such preternatural plagues as God had inflicted on the Egyptians, or the diseases which were peculiar to Egypt, and most frequent in that country, such as the leprosy and other cutaneous diseases."
Somewhat more explicitly,the Benson Commentary on Deuteronomy 7:15 says "It seems to refer not only to the plagues of Egypt, but to some other epidemic diseases, which they remembered to have prevailed among the Egyptians, and by which God had chastised them for their national sins."
In both Exodus and Deuteronomy, God is responsible for inflicting disease and can choose whatever terrible affliction to give those who disobey him. Charles-Edward Amory Winslow (The Conquest of Epidemic Disease: A Chapter in the History of Ideas, pages 36-37) says:
Among the Semitic peoples, the concept of disease as punishment for sin reaches its apogee ... In sharpest contrast with the prevailing demonology of the New Testament, as well as with Babylonian and Persian ideology, the overwhelming mass of references to disease in the Old Testament, from beginning to end, involve the higher concept of punishment for sin and the nobler gospel of righteousness as its only remedy.