He had a relationship with a girl for two years, can he propose marriage to her?

The relationship mentioned in the question is a forbidden relationship and an obvious sin. It is not permissible for a man to form a relationship with a non-mahram woman where they become friends and go out together, because that involves doing things which Allaah has forbidden, namely looking, touching, being alone together and talking softly. Such relationships are not free of these things. It is one of the tricks of the Shaytaan to make such relationships attractive to people, so that a person feels his worries have been relieved or reduced because of this relationship. This is obvious from your words “I turned to” … “her sincerity” “I will never forget” etc. … In addition to your denial of having done anything wrong despite the fact that what has happened is a kind of forbidden relationship, even if we assume that it has not reached the level of zina. 

What you have to do is to repent to Allaah and to be convinced that what you have done is wrong. You have to give up this evil action and regret what has happened in the past. Remember the words of the Prophet (peace and blessings of Allaah be upon him) to the one who came to him and asked him for permission to commit zina: “Would you like it for your mother?” He said: “No, by Allaah, may Allaah cause me to be sacrificed for you.” He said: “And people do not like it for their mothers.” He said: “Would you like it for your daughter?” He said: “No, by Allaah, O Messenger of Allaah, may Allaah cause me to be sacrificed for you.” He said: “And people do not like it for their daughters.” He said: “Would you like it for your sister?” He said: “No, by Allaah, may Allaah cause me to be sacrificed for you.” He said: “And people do not like it for their sisters.” He said, “Would you like it for your paternal aunt?” He said: “No, by Allaah, may Allaah cause me to be sacrificed for you.” He said: “And people do not like it for their paternal aunts.” He said, “Would you like it for your maternal aunt?” He said: “No, by Allaah, may Allaah cause me to be sacrificed for you.” He said: “And people do not like it for their maternal aunts.” 

Narrated by Ahmad, 22265; classed as saheeh by Shaykh Shu’ayb al-Arnaa’oot in Tahqeeq al-Musnad. 

With regard to proposing marriage to her, there is no reason why you should not, after establishing that she is righteous and that she regularly fulfils the duties required in Islam, and that she keeps away from haraam things, and that she has repented from this sinful relationship. 

And Allaah knows best.