Hello and welcome! I am an attorney who will try my very best to help with your situation or get you to someone who can. There may be a slight delay in my responses as I research statutes or ordinances and type out an answer or reply, but rest assured, I am working on your question.
I am very sorry you are dealing with this situation. I am not sure why he feels entitled to the equity in the house merely because he was really a guarantor of the property. If he did not pay anything toward the payments, property taxes, etc... he really does not have an argument as to having equity in the house.
Unfortunately however you can not compel him to remove himself from the deed. You could seek to partition the property and force the sale. In this event a judge could award equity be split but again that is a weak argument on his behalf and likely the house would be sold and he receive no proceeds. The problem is you are now out legal fees and costs along with real estate commissions.
Rather then walk away if he continues to be unreasonable, an option would be to sell the house and have 1/2 of the net proceeds placed in escrow and then have a judge or arbitrator decide if he should get any of the funds. In other words file a claim against the escrow account or him personally for the amount he thinks you owe him.
The other option even though it would not sit well with me but would get rid of the problem is look at what 50% of the net proceeds after costs would be if you were to sell the house and pay him that out of the refinance or sign a promissory note for the amount to pay him over time.
I think he has a weak claim based on the facts presented but he holds a strong hand being a part owner.
The above is assuming the deed is held as joint tenancy with right of survivorship. You will want to check the current deed to see how title is held. If it is not and held as tenancy by the entirety, then you can actually sell or refinance only your 1/2 interest in the house. Now the practicality of this is not ideal. No one is likely to buy 1/2 of a house but it is something that you may be able to lean on him a bit letting him know that you can sell the house on the market as to 1/2 of it and he can own the house with a complete stranger.
I hope this helps and clarifies. Please use the SEND or REPLY button to keep chatting, or please RATE when finished. You may always ask follow ups at no charge after rating. Kindly rate my answer as one of TOP THREE FACES/STARS and then SUBMIT, as this is how experts get credit for our time. Rating my answer the bottom two faces/stars (or failing to submit the rating) does not give me credit and reflects poorly on me, even if my answer is correct. I work very hard to formulate an informative and honest answer for you; please reciprocate my good faith with a positive rating.