How would you write an ethics approval for a study conducted on human data, when the dataset is publicly available?
In general you wouldn't need ethics approval to use this data set; it would be covered by the ethics statement from the original experiment. Presumably the original data collectors had one, even if it's not referenced; if you want to be extra-careful, you should try to find the peer-reviewed publication associated with the data set, and look there for the ethics statement.
More generally, it's hard to see how you could be causing harm by using a public, already collected data set; you didn't have any interaction at all with the subjects! The exception to this would be if you had reason to believe that the original data set might not have been ethically collected, in which there would be an argument for not using the data even though it has already been collected. Practically speaking, though, if these are data collected in the last few decades, it's likely that the researchers were required to undergo ethics approval before data collection.
There are lots of edge cases here:

data that were unethically collected, either decades ago (Tuskegee/Nazis/Stanford Prison study/Milgrom obedience studies etc.) or more recent studies that slipped through the cracks of the modern research-ethics framework
data that are insufficiently anonymized
data that are OK in the jurisdiction where they were originally collected but not in yours

If possible, you should be careful and follow @DianaPetitti's advice by checking with your university's research ethics board; hopefully they will tell you you're good to go. (Although I absolutely understand why it is so, ethics boards are often cautious to the point of [researcher] frustration even when the probability of harm is almost vanishingly small.)
Disclaimer: I am not an ethicist and have not been involved in dealing with ethics statements with the exception of a few surveys and student evaluation exercises.

Tracking the source of the data back we get to Greenwald, S.D. (1990). Improved detection and classification of arrhythmias in noise-corrupted electrocardiograms using contextual information. Thesis. Massachusetts Institute of Technology.
https://dspace.mit.edu/handle/1721.1/29206

We established a test database of 35 half-hour ECG records ...
[Some data] are from the MIT/BIH Arrhythmia Database ...
[Some other data] are from the AHA Arrhythmia Database ...
Tapes in the 800 series were collected for this research from long-term Holter records archived in the Beth Israel Hospital Arrhythmia Laboratory ...

This is admittedly a little disappointing; there's no evidence that informed consent was obtained, although from a practical standpoint it seems incredibly unlikely that these records could either be personally identified, or that any significant harm was caused in their collection. (This describes the data that are first presented in the thesis — for complete diligence you'd have to go back farther to the sources of the other two arrhythmia databases listed here and see what they say about sample collection ..)