"Pediatry" may be an issue, as this is not a term commonly used. You may mean "podiatry", which is the treatment of the feet and their ailments. Alternatively, you may mean "pediatrics", the medical treatment of children. The rest is fine.

It should say Doctor of Pediatrics. Also, the date is not written correctly, there is a comma missing before the year. First, you have to decide whether you are writing it in British or American format. If you go with American, the way you wrote it now, it can be: June 17, 2013 or June the Seventeenth, 2013. If you want the British format, then your best choices are: the Seventeenth of June, 2013 or 17th June 2013. Diplomas are often written with dates spelled out.