Yes! In fact, if he refuses a DNA test, he can be found to be the legal father by the mere refusal to take the test. If you cannot afford a private lawyer, you really need to contact the Los Angeles Department of Child Support Services. I will admit that there will be a large stack of paperwork for you to fill out, but once you do, they will do all the paperwork necessary for you to establish paternity and to get a child support order (at no cost, although some cases do collect a $25 fee over the course of a year). The clock is ticking. Pursuant to California Family Code section 4009, they can only get child support back to the date the appropriate papers are filed, so time is of the essence. This is what they do. Please check out the website....www.childsup.ca.gov.
I've worked as an attorney for Child Support Services for many years (albeit not in Los Angeles County), and we handle cases like yours every day. The father of your child deserves an appropriate DNA test, and we will offer that to him. If he refuses, we will not hesitate to get an order that he comply. If he refuses that, we will get an order declaring him to be the father based on his failure to comply. Trust me, your ex is not the first to try this. The laws in this state have been carefully designed to prevent abuse by the type of intransigence you describe.
Your child deserves financial input from both parents. Don't delay.
Yes, you may file a paternity action either through a private family law action or through Department of Child Support Services. In either case, he will be required to submit to a DNA test. If he fails to follow the court order and submit to the testing, the court will order that he is the father and he will be required to pay child support, even if he is found not to be the biological father of the child.
The response above does not form an attorney-client relationship. This answer may or may not apply to you and should not be relied upon as legal advice.