can you stop the sell of your home if a child support lien shows up after your under contract?

AGAIN, yes

No, you pay the lien out of the sale price, there is no way the contract would let you out for that.

If you did not disclose the lien initially, yes.

Depends on the law where you are, which you do not tell us.

Discovering an unknown lien is not a valid reason for the seller to back out of a contract because it was not a contingency of the sale.

You can always breach the contract, but to do so would put you in jeopardy of being sued for big money. The buyer could not only sue you for the value of the house, but the future value of the house. Rarely does it get taken to that extreme, but it is possible. Even if you have to secure a separate loan to pay off the lien (if necessary) to provide clear title, its in your legal best interest to do so.

As the seller, you can not technically "stop the sale".

As a practical matter, the buyer and/or the buyer's lender will not take title to the property as long as there is a lien on it.

Your contract probably requires that your property is unencumbered.

What usually happens when there is a lien, is that the lien is paid off simultaneously with the sale at closing.

Until the sale is closed either party can back out of the sale. The consequences of doing so should be spelled out in the offer and acceptance. If the buyer has spent money following your acceptance of the offer, they would a valid claim against you to recover those expenses (think inspection reports etc)

Omg

You can't sell the house with the lien on it unless the buyer is stupid. Yes the sale can and most like will be stopped until the lien is taken care of.

If there's enough equity in the house to clear the lien, I don't see how the lien itself would stop the sale. Liens are cleared as part of closing settlement all the time.

You can always refuse to sell, but they can sue you for breach of contract.

If the buyer requests any repairs as a result of the inspection, you can refuse. They might exercise their inspection contingency to terminate the contract at that point in time.