Can I go through security at US airports without being physically touched by an official?

On an unpredictable basis I would say we have been touched roughly 1 time in 20. Beyond that we have drawn searches for things that we had some control over:

1) Two layers of clothing will likely prompt a search. A couple of times flying out of a cold place I was wearing thermal underwear–which has always drawn a search.

2) Large areas of solid or quasi-solid material in your carry-on baggage will confuse the x-ray people and generally lead to a search. Avoid anything that’s going to look like a blob on the x-ray.

3) On my last flight I got a full pat-down due to medical liquids–something that has never happened in the past. I’m not sure if it’s changed procedures or that I drew a guy who didn’t really know what he was doing (kept asking other agents for guidance.) Previously it’s always just been their test strip waved over the bottle (I’ve never been carrying anything that had to remain sterile.)

This has only been the security people when boarding an airplane. Never once have we been touched by the customs and immigration people other than incidental contact while handing documents one way or the other.

There are also the airports in Europe. You might get a security screening at the departure airport in Europe. They are not all equipped with body scanners.

It also depends on the specific airport. Security screening in London (UK) is quite tight, while in Thessaloniki (Greece), you might be lucky and walk just through without screening (especially when you arrive late at the gate). However, it might just happen that even in those more “relaxed” locations, security procedures can be changed without prior notice. Sometimes, random people are picked for a more thorough screening.

At some airports with the tighter security procedures, you are given the option to remove parts of your clothing (like shoes or the belt) so they can be inspected without someone getting too close to you. Belts and shoes are often touched by officials as they are assumed to be good places to conceal illegal items.

if you carry medical equipment on your body (an insuline pump, a CGM, a heart monitor or such things), this equipment might receive additional attention. Some insuline pumps can be detached, others will need to be replaced when removed from the body. CGM sensors will also typically be unusable once removed and need to be replaced as well. If you insist that these devices remain stuck to your body, security officials might want to touch them (not necessarily with their hands, but with a “broom” to collect chemical traces from the surface of this equipment).

If the concern is Haphephobia, then yes, you are likely to pass through all screening without being touched. However, be aware that this is not guaranteed.

The biggest risk is when entering the terminal because if you set of any alarm, either in the mag or scanner, it will have to be resolved, usually by a pat-down of the suspected area.

The reduce this possibility, be sure to wear no metal or dense items, including jewelry, while passing through TSA line. And, be sure your clothes are dry as water spots confuse the image processor.

You would only be touched by CBP if sent to Secondary Screening and they suspect you may be concealing something on your person.

You will be physically searched if drugs are suspected and possibly for currency. Much, much less so for an Agriculture or Duty violation.

So, carry no food products or illicit drugs through Customs. Nothing that would give the detector dogs reason to even look at you.

Country of origin for the arriving flight is a factor in screening methods due to the prevalence of drug trafficking from certain regions.

TSA uses advanced imaging technology to safely screen passengers at the checkpoint. Advanced imaging technology uses automated target recognition software that eliminates passenger-specific images and instead auto-detects potential threats by indicating their location on a generic outline of a person.

The full-body scanners used in all airports since May 2013 are called “millimeter wave” machines, which bounce electromagnetic waves off the traveler to provide an animated image where a suspicious item might be located. The TSA no longer uses the backscatter scanners that produced near-naked images of travelers.

Passengers may decline, in some instances, and opt for a physical screen, in which an officer of the same gender does a pat down search, and which, on request, can be done in a private area away from public view.

“Generally, passengers undergoing screening will still have the option to decline a (full-body) screening in favor of physical screening,” said Bruce Anderson, a TSA spokesman. “However, some passengers will be required to undergo (full-body) screening if warranted by security considerations in order to safeguard transportation security.”

And, as reference, TSA has a video where you can watch both methods in operation.

Most travellers will not be physically screened however there’s no certainty alas. The TSA prefers what they call "Advanced Imaging Technologies" (AIT) compared to what they call "physical screening". However, if the AIT produces an alarm you will be get the "physical screening" as you can find in the Privacy Impact Assessment Update for
TSA Advanced Imaging Technology. And to make things worse, recently they have added that now the "front of the hand may be needed" to the Security Screening page:

A pat-down may include inspection of the head, neck, arms, torso, legs, and feet. This includes head coverings and sensitive areas such as breasts, groin, and the buttocks. You may be required to adjust clothing during the pat-down. The officer will advise you of the procedure to help you anticipate any actions before you feel them. Pat-downs require sufficient pressure to ensure detection.

TSA officers use the back of the hands for pat-downs over sensitive areas of the body. In limited cases, additional screening involving a sensitive area pat-down with the front of the hand may be needed to determine that a threat does not exist.

You will receive a pat-down by an officer of the same gender.