Yes, character as in when candidates promise to endorse the RNC nominee but change their mind
Considering that Trump attacked Cruz's wife and his father during the primaries I am inclined the think that would release him from his promise. My reasoning for this statement is as follows. Ephesians 5:25-28 says:
25 Husbands, love your wives, as Christ loved the church and gave himself up for her, 26 that he might sanctify her, having cleansed her by the washing of water with the word, 27 so that he might present the church to himself in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish. 28 In the same way husbands should love their wives as their own bodies. He who loves his wife loves himself.
How would Cruz be showing his love for his wife by endorsing the man that attacked her? That would not be a loving action towards his wife.
Scripture also says in Exodus 20:12
Honor your father and your mother, that your days may be long in the land that the Lord your God is giving you.
How would endorsing a man you hinted that his father was involved in a presidential assassination be honoring to his father.
Finally, Ephesians 5:11 says
Take no part in the unfruitful works of darkness, but instead expose them.
Trump has exposed that there are many areas of darkness in his character. By supporting Trump Cruz would be breaking this commandment from God as well.
To recap, If Cruz was to support Trump he would be breaking the command to love his wife as Christ loves the church, he would be breaking the command to honor his father by joining to a man who had slandered him, and he would be breaking the command to not take part in the unfruitful works of darkness.