If a person swears an oath silently, is his oath binding?

One of the conditions of an oath being binding is that the one who swears it should speak his oath, i.e., by moving his tongue. If he does not move his tongue, then the oath is not binding, because of the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “Verily Allah has pardoned my ummah for what crosses their minds, so long as they do not do it or speak of it.” 

Qataadah said: If the word of divorce crosses his mind, then it does not mean anything.

Narrated by al-Bukhaari (5269) 

It says in Fataawa al-Lajnah ad-Daa’imah (vol. 1, 20/214):

If the situation is as you describe, that you did not utter the word of divorce out loud or silently, and did not swear an oath to that effect – rather it was only something that crossed your mind – then this does not have any impact, and no consequences result from it, neither divorce nor expiation. End quote. 

For more information, please see the answers to questions no. 114871 and 34164 

Moving the lips is more significant than moving the tongue, because moving them can hardly happen except by moving the tongue. 

Imam Ahmad narrated in his Musnad (10968) from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Verily Allah, may He be glorified and exalted, said: ‘I am with My slave when he remembers Me and moves his lips in mention of Me.’” Classed as saheeh by al-Albaani and the commentators on al-Musnad. 

Imam al-Bukhaari (may Allah have mercy on him) compiled a chapter on the verse: “Move not your tongue concerning (the Quran, O Muhammad (blessings and peace of Allah be upon him)) to make haste therewith” [al-Qiyaamah 75:16], and quoted this hadith qudsi in the chapter. 

Al-Mullah ‘Ali al-Qaari said in Mirqaat al-Mafaateeh Sharh Mishkaat al-Masaabeeh (4/1560):

“when he remembers Me” i.e., in his heart and verbally

“and moves his lips in mention of Me.” At-Teebi said: Referring to the lips rather than the tongue is emphasizing the mentioning of Allah.  End quote.

This indicates that moving the lips implies moving the tongue, and there is a greater emphasis on mentioning Allah. 

Based on that, this oath that you swore is a binding oath, which you must fulfil, and you must offer expiation if you broke the oath. 

And Allah knows best.