Why didn't Mordechai divorce Esther?

 One of the most dramatic moments in the Megila happens when Mordechai tells Esther to take the initiative and go to the King with the hope of persuading him to help save the Jewish people

 Esther (chapter 4)

13. Then Mordecai commanded to answer Esther, Think not yourself that in the king’s palace you shall escape, any more than all the Jews.
14. For if you remain silent at this time, then shall relief and deliverance arise to the Jews from another place; but you and your father’s house shall be destroyed. And who knows whether you have not come to the kingdom for such a time as this?
15. Then Esther bade them return Mordecai this answer,
16. Go, gather together all the Jews who are present in Shushan, and fast for me, and neither eat nor drink three days, night or day; I also and my girls will fast likewise; and so will I go to the king, though it is against the law; and if I perish, I perish.

The gemorah Megila (15a)R. Abba said: It will not be she said according to the custom of every other day. Till now I have associated with Ahasuerus under compulsion, but now I will do so of my own will.  And if I perish, I perish. As I am lost to my father's house so I shall be lost to thee.

The gemorah is saying that Esther was married to Mordechai and up to this point she was vied as being raped by the king and thus was still allowed to remain married to Mordechai.  But now 

Mordechai is ordering her to go of her own free will to the king and commit adultery thus henceforth she will be forbidden to Mordechai. Tosfos asks why Moredchai doesn't simply divorce her 

He answers because the king might find out about it since all gittin require witnesses. the Nodah beyehuda says he could divorce her without witnesses.

תוספות מסכת מגילה דף טו עמוד א
כשם שאבדתי מבית אבא כך אבדתי ממך - וא"ת אמאי לא היה מגרשה ותהא מותרת להחזירנה י"ל לפי שכל מעשה הגט הוא ע"פ עדים והיה ירא פן יתפרסם הדבר למלכות.