@JoeStrazzere has described what you need to do right this minute to salvage your relationship. But I think you also need to learn to avoid putting yourself in such positions in the future. So I want to discuss how to disagree technically with someone who is your organizational superior and how to get your ideas accepted.
First what not to do (I'm not saying you did these things, but these are things to watch out for in the future):
Never embarrass your boss publicly with an argument. Yes, you can disagree in a meeting as long as it is just your team (never do it if there are people from higher levels there) and as long as the boss has opened up the issue for discussion.
Don't continue to argue after the decision has been made. The time to bring up a discussion of a technical issue is before the decision is made. Once a decision is made, you implement it to the best of your abilities without whining. Being a sore loser is professional suicide. Helping the boss implement his solution will get you respect points that you need the next time you want to bring up a technical change.
Don't argue, discuss. If you start to lose your temper, take a bathroom break to calm down.
Don't disagree on everything. You will get a reputation as unpleasant to deal with and people will just tune you out.
Now as to how to get your technical ideas the consideration they deserve and have a chance to get them implemented:
First gain professional credibility by being the person who delivers the work and does it well. The new employee and/or junior ones have less credibility. To have professional credibility, you have to have accomplishments that your boss has directly seen. (You may think this isn't fair, but life isn't fair, so deal.)
Make your presentation as early as possible in the design process. Be aware that if you come in to a job in the middle, there are decisions that you may disagree with at that point, but it is now too late to change them.
Describe your ideas in terms of meeting his business needs not just in technical terms. It might be nice to do XYZ, but if it is a new technology that we don't know, it might add time we don't have in order to meet the deadline, the new stuff might be harder to integrate with existing legacy code. Tech leads and PMs have to keep all those business considerations in mind when making design choices. There are no purely technical design issues. Business needs and politics are part of every single decision and you need to not only get used to that, but learn to use them to your advantage. Learn how to do cost-benefit analysis.
Be realistic. We all like to play with new toys, but the business needs are more important than your personal desire to try out something new. Suggesting a rewrite of a ten-year old application so that you can use this exciting new technology is never going to go over well. While you may not like the legacy stuff, if it currently works, changing it to use some fancy new technology is a very risky thing for a business to do. It is expensive and there is often no benefit to the users. It risks breaking things that currently work. When working in legacy systems, think more in terms of suggesting incremental improvements. And do try to remember that your boss probably wrote a good portion of that legacy system and he is proud of it.
Be professional in your presentation and do not patronize the person who is ultimately responsible for the decision. Telling the boss you think he or she is stupid either in actual words or in body language and tone of voice is a sure way to be discounted in any decision making. Leave the "I'm better than you" attitude at home.
Solicit ideas from others and ask for some help from others in refining your presentation before making it to the decision maker. This helps you get allies which will help you get ideas taken seriously.
Choose your ground carefully. To get accepted, an idea needs to be worth doing and the timing needs to be good. When the boss has been having a bad day, you can go in with a solution to fix whatever the problem that day was. But don't go in to discuss some unrelated issue that you want done. You will have more luck when he is in a good mood unless you are suggesting a fix to his immediate problem. Also bringing up something that will take time to discuss when he is getting ready to leave for the day or headed to a meeting in five minutes is not great timing either unless the issue is genuinely urgent.
If you are new to the organization or profession, start small with something that is likely to be an easy sell. Building a reputation for having good ideas will help you professionally. So having some small successes to build on will help you. Once you have that reputation though, then concentrate on the important stuff.
If your idea gets accepted, then make sure you do everything you can to make it work. A failed idea won't kill your career, but a string of them will. The more successful your ideas are, the more credibility you have the next time you want to bring up a different way to do something.