It helps to think about it like laundry. In a washing machine one might assume dirt, oil, and debris comes loose through agitation alone. However, wouldn't the clothing act like a thick, giant filter keeping the oil, dirt, and debris on clothing when the water is drained? If not, wouldn't running a a single rinse cycle clean your clothes?
First you have to understand how soap works. To accomplish that you need to understand what hydrophyllic and hydrophobic compounds are.
Nearly all compounds fall into one of two categories: hydrophilic ('water-loving') and hydrophobic ('water-hating'). Water and anything that will mix with water are hydrophilic. Oil and anything that will mix with oil are hydrophobic. When water and oil are mixed they separate. Hydrophilic and hydrophobic compounds just don't mix.(What is Soap)
The cleansing action of soap is determined by its polar and non-polar structures in conjunction with an application of solubility principles. The long hydrocarbon chain is non-polar and hydrophobic (repelled by water). The "salt" end of the soap molecule is ionic and hydrophilic (water soluble).(What is Soap)
When grease or oil (non-polar hydrocarbons) are mixed with a soap- water solution, the soap molecules work as a bridge between polar water molecules and non-polar oil molecules. Since soap molecules have both properties of non-polar and polar molecules the soap can act as an emulsifier. An emulsifier is capable of dispersing one liquid into another immiscible liquid. This means that while oil (which attracts dirt) doesn't naturally mix with water, soap can suspend oil/dirt in such a way that it can be removed. The soap will form micelles (see below) and trap the fats within the micelle. Since the micelle is soluble in water, it can easily be washed away. - (What is Soap)
The purpose of cleaning dishes is to remove debris, sugars, salt, oils, protiens and fats which would otherwise serve as a medium for bacterial growth. Soap is a surfacant and attaches to the water. So, as @Anno2001 mentioned in his comment without knowing the reason, the action that allows dirty soap mixed with water to slide off an object via gravity is due to polarity,and the hydrophyllic / hyrophobic properties of soap and water.
So, technically no, it isn't unhealthy. Even if detergent with known toxic chemicals were used, it still wouldn't adhere to the surface and be consumed later.
Bringing it back to the washing machine analogy, a measure is given for the proper amount of detergent to use. The same knowledge applies to dish soap. However, common sense needs to be applied, since soap to water ratios or instructions aren't on given for dish soap.* Experience and tactility tells us when we've used too much soap.
This fact can be peer-reviewed and replicated by enough people using a search engine "how to use dish soap" or "what are the instructions for dish soap."